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A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed $$q \ge 2$$ q ≥ 2 , we give an $$\mathscr {O}^{\star }(q^{{\mathsf {tw}}})$$ O ⋆ ( q tw ) -time algorithm when the input graph is given together with one of its tree decompositions of width $${\mathsf {tw}} $$ tw . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is $$\mathsf {XP}$$ XP parameterized by clique-width.

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration. We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied. We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases., LIPIcs, Vol. 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), pages 6:1-6:18

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$, and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbb{Q}$-rank-width, rank-width, and maximum induced matching width of a given decomposition. Our approach simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic \sf NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [B. Bui-Xuan, J. A. Telle, and M. Vatshelle, Theoret. Comput. Sci., (2013), pp. 66--76] and the rank-based approach introduced in [H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof, Inform. and Comput., 243 (2015), pp. 86--111]. The results we obtain highlight the importance of the $d$-neighbor equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for ${\sf W}[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$, and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbb{Q}$-rank-width, and the rank-width of the input graph. publishedVersion

In this paper we describe our system for automatically extracting "correct" programs from proofs using a development for the Curry-Howard process. Although program extraction has been developed by many authors (see [5,?,?]), our system has a number of novel features designed to make it very easy to use and as close as possible to ordinary mathematical terminology and practice. These features include 1. the use of Henkin's technique [6] to reduce higher-order logic to many-sorted (first-order) logic 2. the free use of new rules for induction subject to certain conditions 3. the extensive use of previously programmed (primitive) recursive functions. 4. the use of templates to make the reasoning much closer to normal mathematical proffs. 5. an extension of the technique of the use of Harrop formulae to classically true formulae (cf. the footnote on p.101 in Kreisel [9]); As an example of our system we give a constructive proof of the well-known theorem that every graph of even parity, which is non-trivial in the sense that it does not consist of isolated vertices, has a cycle. Given such a graph as input, the extracted program produces a cycle as promised.

We prove that ( 4 K 1 , C 4 , C 5 , C 7 ) -free graphs that are not chordal have unbounded clique-width. This disproves a conjecture from Fraser et al. (2017).

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. We continue a recent study into the clique-width of $(H_1,H_2)$-free graphs and present three new classes of $(H_1,H_2)$-free graphs of bounded clique-width and one of unbounded clique-width. The four new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of clique-width for the first three cases we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs. We extend our proof of unboundedness for the fourth case to show that Graph Isomorphism is Graph Isomorphism-complete on the same graph class. We also show the implications of our results for the computational complexity of the Colouring problem restricted to $(H_1,H_2)$-free graphs., Comment: 22 Pages, 2 figures, An extended abstract of this paper will appear in the proceedings of LATA 2015 (LNCS vol. 8977)

Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}\cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$ or worse.

The 0–1-knapsack problem is a well-known NP-hard problem in combinatorial optimization. We consider the extensions to the knapsack problem with conflict graph (KCG) and the knapsack problem with forcing graph (KFG). KCG has first been introduced by Yamada et al. and represents incompatibilities between items of the knapsack instance. KFG has been introduced by Pferschy and Schauer and represents the necessity of items for other items. Within this paper we provide pseudo-polynomial solutions for KCG and KFG with co-graphs as conflict and forcing graphs and extend these solutions to conflict and forcing graphs of bounded clique-width. Our solutions are based on dynamic programming using the tree-structure representing the conflict graph and the forcing graph. Further we conclude fully polynomial time approximation schemes for KCG on conflict graphs of bounded clique-width and KFG on forcing graphs of bounded clique-width. This generalizes the known results for conflict graphs and forcing graphs of bounded tree-width of Pferschy and Schauer.

Finding a diagonal matrix congruent to $A - cI$ for constants $c$, where $A$ is the adjacency matrix of a graph $G$ allows us to quickly tell the number of eigenvalues in a given interval. If $G$ has clique-width $k$ and a corresponding $k$-expression is known, then diagonalization can be done in time $O(\text{poly}(k) n)$ where $n$ is the order of $G$., Comment: 25 pages

M AX -C UT , E DGE D OMINATING S ET , G RAPH C OLORING , and H AMILTONIAN C YCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO 2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO 1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g ( k ) n f ( k ) on graphs of clique-width k . Fomin et al. (2010) showed that the running times cannot be improved to g ( k ) n O (1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f ( k ) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for E DGE D OMINATING S ET and M AX -C UT to n O ( k ) , and proved that these problems cannot be solved in time g ( k ) n o ( k ) unless ETH fails. Thus, prior to this work, E DGE D OMINATING S ET and M AX -C UT were known to have tight n Θ ( k ) algorithmic upper and lower bounds. In this article, we provide lower bounds for H AMILTONIAN C YCLE and G RAPH C OLORING . For H AMILTONIAN C YCLE , our lower bound g ( k ) n o ( k ) matches asymptotically the recent upper bound n O ( k ) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n Θ( k ) bounds for E DGE D OMINATING S ET , M AX -C UT , and H AMILTONIAN C YCLE , the G RAPH C OLORING problem has an upper bound of n O (2 k ) and a lower bound of merely n o (√ [4] k ) (implicit from the W[1]-hardness proof). In this article, we close the gap for G RAPH C OLORING by proving a lower bound of n 2 o ( k ) . This shows that G RAPH C OLORING behaves qualitatively different from the other three problems. To the best of our knowledge, G RAPH C OLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n .

Recently, there have been found relations between the algebraic co-rank and the zero forcing number along with the minimum rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank. This sheds some light on the conjecture that the real minimum rank is bounded from above by the real algebraic co-rank.

In this paper, we show that for a fixed k, there is an NC algorithm that separates the graphs of rank-width at most k from those with rank-width at least, by Bireswar Das, Anirban Dasgupta, Murali Krishna Enduri, and Vinod I. Reddy, 2018-08-10 12:16:50, https://doi.org/10.1016/j.ipl.2018.07.007

Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability. In this paper we design a succinct data structure for graphs on $n$ vertices whose clique-width is at most $k \leq \epsilon \sqrt{\log n / \log \log n}$ for some constant $0 , along with supporting degree, adjacency, neighborhood queries efficiently. This resolves an open problem of Kamali (Algorithmica-2018).

Given an undirected graph, a conflict-free coloring (CFON*) is an assignment of colors to a subset of the vertices of the graph such that for every vertex there exists a color that is assigned to exactly one vertex in its open neighborhood. The minimum number of colors required for such a coloring is called the conflict-free chromatic number. The decision version of the CFON* problem is NP-complete even on planar graphs.

This, the last chapter of the book, might be called ‘beyond line graphs’ because most of the graphs under consideration are not in fact line graphs. Recall that Krausz’s characterization theorem was this: A graph is a line graph if and only if its edges can be partitioned into complete subgraphs in such a way that each vertex is in just one or two of the subgraphs. Here, we extend this idea and define the Krausz dimension of a graph G to be the minimum value of d for which the edge set of G can be partitioned into complete subgraphs in such a way that each vertex is in at most d of the subgraphs. This turns out to be an NP-complete parameter, and so our focus is on bounds and critical graphs.

An acyclic r-coloring of a directed graph \(G=(V,E)\) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the \(\text {W}[1]\)-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width. From a parameterized point of view our algorithm shows that the Dichromatic Number problem is in \(\text {XP}\) when parameterized by directed clique-width and extends the only known structural parameterization by directed modular width for this problem. Furthermore, we apply defineability within monadic second order logic in order to show that Dichromatic Number problem is in \(\text {FPT}\) when parameterized by the directed clique-width and r. For directed co-graphs, which is a class of digraphs of directed clique-width 2, we even show a linear time solution for computing the dichromatic number.

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify ���-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of ���-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can approximate every possible ���-branchwidth, providing a unifying parameterized framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions., LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 63:1-63:23

In this paper, we present an algorithm to approximate the clique-width of a graph. The proposed approach is based on computing the shortest paths between pairs of vertices. We experimentally show that our proposal approximates the clique-width of simple graphs in polynomial time, while other methods that calculate it in an exact way, transform the problem to SAT, that is well-known as NP-Complete.

Representations of graphs are a way of encoding the structure of a graph by using other discrete structures. The object of a representation of a graph is to encode its structure efficiently. Typically a graph can be encoded by an n× n adjacency matrix. It is possible, however, to encode graphs much more efficiently using other representation schemes. This work considers tree representations of graphs. A tree representation of a target graph G is an assignment of subtrees of a host tree to the vertices of G in such a way that if uv ∈ E(G), then the subtree assigned to the vertex u and the subtree assigned to the vertex v have at least t nodes in common. This study considers tree representations such that the host tree comes from the family of subdivided n-stars. The largest such representable asteroidal set is constructed, and a lower bound on the length of the longest cycle representable on this family of host tree is also discovered. Next we move to a different area of study. The study of closure systems and closure spaces is a relatively new direction in mathematics. Jamison writes in his new text that ‘the notion of closure is pervasive throughout mathematics’. Surely, closure and closed sets can be discussed in almost any mathematical setting. It has been shown by Pfaltz in 1995 that for any finite ground set S, with |S| = n such that n ≥ 10, there are n unique closure operators on S. In topology, the separation properties provide criteria for categorizing topological spaces. While not all closure spaces are topological spaces, we may still explore whether the separation properties hold under certain conditions. This work defines a class of closure operators on the integers and investigates the conditions for which the resulting closure space satisfies the different definitions of separability.

Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Reddy ’18 about parameterization by the distance to threshold graphs by showing that MHE is \(\mathrm {NP}\)-complete on threshold graphs. Hence, it is not even in \(\mathrm {XP}\) when parameterized by clique-width, since threshold graphs have clique-width at most two. As a complement for this result we provide a \(n^{\mathcal {O}(\ell \cdot \mathrm {cw})}\) algorithm for MHE, where \(\ell \) is the number of colors and \(\mathrm {cw}\) is the clique-width of the input graph. We also construct an \(\mathrm {FPT}\) algorithm for MHV with running time \(\mathcal {O}^*((\ell +1)^{\mathcal {O}(\mathrm {cw})})\), where \(\ell \) is the number of colors in the input. Additionally, we show \(\mathcal {O}(\ell n^2)\) algorithm for MHV on interval graphs.

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class \(\mathcal{G}\) if they are so on the atoms (graphs with no clique cut-set) of \(\mathcal{G}\). Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is \((H_1,H_2)\)-free if it is both \(H_1\)-free and \(H_2\)-free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for \((H_1,H_2)\)-free graphs, as evidenced by one known example. We prove the existence of another such pair \((H_1,H_2)\) and classify the boundedness of clique-width on \((H_1,H_2)\)-free atoms for all but 18 cases.

Given a graph \(G=(V,E)\), the vertex expansion of a set \(S\subset V\) is defined as \( \varPhi ^V(S)=\frac{|N(S)|}{|S|}. \) In the Small Set Vertex Expansion (SSVE) problem, we are given a graph \(G=(V,E)\) and a positive integer \(k\le \frac{|V(G)|}{2}\), the goal is to return a set \(S\subset V(G)\) of k nodes minimizing the vertex expansion \( \varPhi ^V(S)=\frac{|N(S)|}{k}\); equivalently minimizing |N(S)|. SSVE has not been as well studied as its edge-based counterpart Small Set Expansion (SSE). SSE, and SSVE to a less extend, have been studied due to their connection to other hard problems including the Unique Games Conjecture and Graph Colouring. Using the hardness of Minimum k-Union problem, we prove that Small Set Vertex Expansion problem is NP-complete. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard when parameterized by k, (2) the problem is fixed-parameter tractable (FPT) when parameterized by the neighbourhood diversity nd, and (3) it is fixed-parameter tractable (FPT) when parameterized by treewidth tw of the input graph.

The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. The Balanced Satisfactory Partition problem is a variant of the above problem where the two partite sets are required to have the same cardinality. Both problems are known to be NP-complete. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The three main results of the paper are the following: (1) The Satisfactory Partition problem is polynomial-time solvable for block graphs, (2) The Satisfactory Partition problem and its balanced version can be solved in polynomial time for graphs of bounded clicque-width, and (3) A generalized version of the Satisfactory Partition problem is W[1]-hard when parametrized by treewidth.

Golumbic and Trenk first introduced tolerance graph [7] to generalize some well-known applications of interval graphs. In a tolerance graph, every vertex is represented by an interval and a tolerance such that an edge occurs if and only if the overlap of corresponding intervals is at least as large as the tolerance associated with one of the vertices. Resource allocation sharing tolerate among users and certain scheduling problems can be modeled by tolerance graphs.

Recently, people, devices, processes, and other entities have been more connected than at any other point in history. In general, a graph is a natural, neat, and flexible structure to model the complex relationships, interactions, and interdependencies between objects (Fig. 4.1). In particular, each graph consists of nodes (or vertices) that represent objects and edges (or links) that represent the relationships among the graph nodes. Graphs have been widely used to represent datasets in a wide range of application domains such as social science, astronomy, computational biology, telecommunications, computer networks, semantic Web, protein networks, and many others (Sakr and Pardede, Graph Data Management: Techniques and Applications, 2011 [73]).

We rephrase the Distance labeling problem as a specific uniform variant of the Channel Assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the Distance labeling problem is FPT when parameterized by the neighborhood diversity, the maximum p i and k . This is indeed a more general answer to an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. Finally, we show that the uniform variant of the Channel Assignment problem becomes NP -complete when generalized to graphs of bounded clique width.

A more descriptive but too long title would be : Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute " on the fly " the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quan-tifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.

Does well-quasi-ordering by induced subgraphs imply bounded clique-width for hereditary\ud classes? This question was asked by Daligault, Rao, and Thomass ́e [Well-quasi-order of\ud relabel functions. Order, 27(3) (2010), 301–315]. We answer this question negatively by\ud presenting a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered\ud by the induced subgraph relation. We also show that graphs in our class have at most\ud logarithmic clique-width and that the number of minimal forbidden induced subgraphs for\ud our class is infinite. These results lead to a conjecture relaxing the above question and to a\ud number of related open questions connecting well-quasi-ordering and clique-width.

Drawings of graphs have many applications and they are nowadays well-established tools in computer science in general, and optimization in particular. Project scheduling is one of the many areas in which representation of graphs constitutes an important instrument. The experience shows that the main quality desired for drawings of graphs is readability, and crossing reduction is a fundamental aesthetic criterion to achieve it. Incremental or dynamic graph drawing is an emerging topic in this context, where we seek to preserve the layout of a graph over successive drawings. In this paper, we target the edge crossing reduction in the context of incremental graph drawing. Specifically, we apply a mathematical programming formulation and several heuristic methods based on the tabu search methodology to solve it. In line with the previous paper on this topic, we consider bipartite graphs in our experimentation. The extensive computational experiments with more than 1000 instances show the superiority of our proposals in both, quality and computing time.

This paper introduces a system for measuring the elegance of a graph based on the steps needed to build the graph and on its symmetry structure. The measure is designed to capture the essence of the notion of elegance in mathematics, namely, simplicity and clarity. The term “elegance” is used instead of “aesthetics” to distinguish the measure from those dependent on visual representation of a graph. Elegance is based solely on the abstract properties of a graph. A framework for measurement is defined and applied in a special case.

Graph transduction methods have been widely adopted for label prediction under semi-supervised settings. To alleviate the relevant sensitivity to initial labels and graph construction processes, recent studies have been aiming at developing robust graph transduction techniques. In particular, the graph transduction method via greedy gradient Max-Cut (GGMC) that minimizes a cost function over a continuous classification function and a binary label variable has been successfully applied to a wide range of applications. However, this method predominately relies on the choice of a high-quality single graph representation, often leading to unstable performance due to selection bias. To tackle this major drawback, we leverage an ensemble learning framework into the GGMC method for exploiting the advantage of constructing and combining multiple graphs. As opposed to performing constrained Max-Cut on a single graph, the proposed multiple graph greedy gradient Max-Cut method (MG-GGMC) simultaneously solves the label prediction and the true graph estimation problems. Specifically, the true graph is approximated by a linear combination of a set of constructed graphs. The coefficients of the linear combination are learned automatically by alternately minimizing a unified objective function in an iterative manner. Comparison studies with representative methods across various real-world benchmarks conspicuously demonstrate the efficaciousness and the superiority of the proposed algorithm in standard evaluation metrics.

One of the most famous algorithmic meta-theorems states that every graph property which can be defined in counting monadic second order logic (CMSOL) can be checked in linear time on graphs of bounded treewidth, which is known as Courcelle’s Theorem (Courcelle, 1990). These algorithms are constructed as finite state tree automata and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. In this paper we prove two special cases of this conjecture, first for the class of k -outerplanar graphs, which are known to have treewidth at most 3 k − 1 (Bodlaender, 1998) and for graphs of bounded treewidth without chordless cycles of length at least some constant l . We furthermore show that for a proof of Courcelle’s Conjecture it is sufficient to show that all members of a graph class admit constant width tree decompositions whose bags and edges can be identified with MSOL-predicates. For graph classes that admit MSOL-definable constant width tree decompositions that have bounded degree or allow for a linear ordering of all nodes with the same parent we even give a stronger result: In that case, the counting predicates of CMSOL are not needed.

In this paper, we describe some highlights of the new branch quantitative graph theory and explain its significant different features compared to classical graph theory. The main goal of quantitative graph theory is the structural quantification of information contained in complex networks by employing a measurement approach based on numerical invariants and comparisons. Furthermore, the methods as well as the networks do not need to be deterministic but can be statistic. As such this complements the field of classical graph theory, which is descriptive and deterministic in nature. We provide examples of how quantitative graph theory can be used for novel applications in the context of the overarching concept network science.

Graphs are widely used to model complex data in many applications, such as bioinformatics, chemistry, social networks, pattern recognition. A fundamental and critical query primitive is to efficiently search similar structures in a large collection of graphs. This article mainly studies threshold-based graph similarity search with edit distance constraints. Existing solutions to the problem utilize fixed-size overlapping substructures to generate candidates, and thus become susceptible to large vertex degrees and distance thresholds. In this article, we present a partition-based approach to tackle the problem. By dividing data graphs into variable-size non-overlapping partitions, the edit distance constraint is converted to a graph containment constraint for candidate generation. We develop efficient query processing algorithms based on the novel paradigm. Moreover, candidate-pruning techniques and an improved graph edit distance verification algorithm are developed to boost the performance. In addition, a cost-aware graph partitioning method is devised to optimize the index. Extending the partition-based filtering paradigm, we present a solution to the top- $$k$$ graph similarity search problem, where tailored filtering, look-ahead and computation-sharing strategies are exploited. Using both public real-life and synthetic datasets, extensive experiments demonstrate that our approaches significantly outperform the baseline and its alternatives.

In the past several years, lots of out-of-core graph processing systems are built to process big graph datasets in computer systems with limited main memory. Due to the iterative nature of graph algorithms, most of these systems employ synchronous execution model to organize the computation, i.e., divide the computing into multiple rounds, each of which corresponds to one iteration of the graph algorithm. In order to fully utilize the disk bandwidth, these systems sequentially scan the whole graph dataset at each iteration. However, as the graph dataset under processing may be huge, more iterations generally means larger I/O overheads. Although asynchronous implementation of the synchronous execution model allows message passing within an iteration, the effectiveness is still limited. Since in such model, at most one message is allowed to be passed from one vertex to another. In this paper, we investigate the idea of block updating in the synchronous execution model framework in the out-of-core graph processing systems. With this new model, the system conducts graph algorithm on the loaded subgraph (i.e., block) to its local convergence, and then switches to other subgraphs to continue this process, until global convergence is reached. We implement this new model in GraphChi (the result system is called BlockGraphChi), and propose a companion graph partition method, named as DMLP. By this study, we found that compared with the original execution model of GraphChi: (1) the new model can generally reduce the amount of iterations (and thus the I/O overheads) for graph algorithms, while the extent of reduction depends on the method of graph partitioning and the properties of the algorithms; (2) the new model can dramatically reduce the overall execution time of graph traversal algorithms (by up to 31.4 $$\times $$ ), and better partitioning method leads to better performance; (3) the new model has much smaller effectiveness on improving the overall performance of fix-point algorithms, such as PageRank, due to the increased computational overhead.

With the emerging of large scale complex networks, graph computation, such as community detection, meets new technology challenges of extremely large computational cost. In order to deal with these challenges, the parallelism is becoming necessary. Graph partition is a fundamental problem of parallel computing for big graph data. The challenges of graph partition include large numbers of communications between partitions, extreme replication of vertices, and unbalanced partition. In this paper, we propose a feasible graph partition framework for parallel computing of big graph. The framework is based on an objective optimization problem to reduce the bandwidth, memory and storage cost on condition that the load balance could be guaranteed. In this framework, three greedy graph partition algorithms are proposed. By running the algorithms on the different kinds of graphs, including real-world graphs and synthetic graphs, the experimental results show that our algorithms surpass the state of the art heuristic algorithms. For example, running time is reduced more than 21.56% and the communication cost is decreased by more than 17.90% for weighted graph. The adequate experiments verify that our algorithms are capable of solving the problem of graph partition with different needs.

The notion of clique-width for graphs offers many research topics and has received considerable attention in the past decade. A graph has clique-width k if it can be represented as an algebraic expression on k labels associated with its vertices. Many computationally hard problems can be solved in polynomial time for graphs of bounded clique-width. Interestingly also, many graph families have bounded clique-width. In this paper, we consider the problem of preprocessing a graph of size n and clique-width k to build space-efficient data structures that support basic graph navigation queries. First, by way of a counting argument, which is of interest in its own right, we prove the space requirement of any representation is $$\varOmega (kn)$$Ω(kn). Then we present a navigation oracle which answers adjacency query in constant time and neighborhood queries in constant time per neighbor. This oracle uses O(kn) space (i.e., O(kn) bits). We also present a degree query which reports the degree of each given vertex in $$O(k\log ^*(n))$$O(klogź(n)) time using $$O(kn \log ^*(n))$$O(knlogź(n)) bits.

In this paper, we use a branch decomposition technique to improve approximations to the p-median problem. Starting from a support graph produced either by a combination of heuristics or by linear programming, we use dynamic programming guided by a branch decomposition of that support graph to find the best p-median solution on the support graph. Our results show that when heuristics are used to build the support graph and the support graph has branchwidth at most 7, our algorithm is able to provide a solution of lower cost than any of the heuristic solutions. When linear programming is used to build the support graph and the support graph has branchwidth at most 7, then our algorithm provides better solutions than popular heuristics and is faster than integer programming. Thus, our algorithm is a useful practical tool when support graphs have branchwidth at most 7.

In a wide variety of emerging data-intensive applications, such as social network analysis, Web document clustering, entity resolution, and detection of consistently co-expressed genes in systems biology, the detection of dense subgraphs cliques and k-plex is an essential component. Unfortunately, these problems are NP-Complete and thus computationally intensive at scale — hence there is a need to come up with techniques for distributing the computation across multiple machines such that the computation, which is too time-consuming on a single machine, can be efficiently performed on a machine cluster given that it is large enough. In this paper, we first propose a new approach for maximal clique and k-plex enumeration, which identifies dense subgraphs by binary graph partitioning. Given a connected graph G = ( V , E ) , it has a space complexity of O ( | E | ) and a time complexity of O ( | E | μ ( G ) ) , where μ ( G ) represents the number of different cliques (k-plexes) existing in G . It recursively divides a graph until each task is sufficiently small to be processed in parallel. We then develop parallel solutions and demonstrate how graph partitioning can enable effective load balancing. Finally, we evaluate the performance of the proposed approach on real and synthetic graph data and show that it performs considerably better than existing approaches in both centralized and parallel settings. In the parallel setting, it can achieve the speedups of up to 10x over existing approaches on large graphs. Our parallel algorithms are primarily implemented and evaluated on MapReduce, a popular shared-nothing parallel framework, but can easily generalize to other shared-nothing or shared-memory parallel frameworks. The work presented in this paper is an extension of our preliminary work on the approach of binary graph partitioning for maximal clique enumeration. In this work, we extend the proposed approach to handle maximal k-plex detection as well.

This paper aims at bridging the two important trends in efficient graph cuts in the literature, the one is to decompose a graph into several smaller subgraphs to take the advantage of parallel computation, the other is to reuse the solution of the max-flow problem on a residual graph to boost the efficiency on another similar graph. Our proposed parallel dynamic graph cuts algorithm takes the advantages of both, and is extremely efficient for certain dynamically changing MRF models in computer vision. The performance of our proposed algorithm is validated on two typical dynamic graph cuts problems: the foreground-background segmentation in video, where similar graph cuts problems need to be solved in sequential and GrabCut, where graph cuts are used iteratively.

Automated graph-drawing systems utilize procedures to place vertices and arcs in order to produce graphs with desired properties. Incremental or dynamic procedures are those that preserve key characteristics when updating an existing drawing. These methods are particularly useful in areas such as planning and logistics, where updates are frequent. We propose a procedure based on the scatter search methodology that is adapted to the incremental drawing problem in hierarchical graphs. These drawings can be used to represent any acyclic graph. Comprehensive computational experiments are used to test the efficiency and effectiveness of the proposed procedure.

Graph grammars may be used to formally describe computational systems, modeling the states as graphs and the possible state changes as rules (whose left- and right-hand sides are graphs). The behavior of the system is defined by the application of these rules to the state-graphs. From a practical point of view, the extension of rules to enable description of extra conditions that must be satisfied upon rule application is highly desirable. An example is the specification of negative application conditions, or NACs, that describe situations that prevent the application of a rule. This extension of the basic formalism enhances the expressiveness of rules, generally allowing simpler specifications. Another extension that is fundamental for practical applications is the possibility to use data types, like natural numbers, lists, etc., as attributes of graphical elements (vertices and edges). Attributed graph grammars are well-investigated and used. However, there is a lack of verification techniques for this kind of grammar mainly due to the fact that data types are typically infinite domains, and thus techniques like model checking can not be used directly (without abstraction constructions). The present work provides a theoretical foundation for theorem proving graph grammars with negative application conditions and attributes. This is achieved by generating an event-B model from a graph grammar. Event-B models are composed by sets and axioms to define types, and by states and events to describe behavior. After constructing the event-B model that is semantically equivalent to a graph grammar, properties about reachable states may be proven using the various theorem provers available for event-B in the Rodin platform. This strategy allows the verification of systems with infinite-state spaces without using any kind of approximation.

Labelings that are used in graph decompositions are called Rosa-type labelings. The gamma-labeling of an almost-bipartite graph is a natural generalization of an alpha-labeling of a bipartite graph. It is known that if a bipartite graph G with m edges possesses an alpha-labeling or an almost-bipartite graph G with m edges possesses a gamma-labeling, then the complete graph K_{2mx+1} admits a cyclic G-decomposition. A variation of an alpha-labeling is introduced in this paper by allowing additional vertex labels and some conditions on edge labels and show that whenever a bipartite graph G admits such a labeling, then there exists a supergraph H of G such that H is almost-bipartite and H has a gamma-labeling.

An alliance in a graph is a set of vertices that is either safe under attacks from the neighborhood (defensive), capable of attacking its neighbors (offensive), or simultaneously defensive and offensive (powerful). An alliance is global if all nonmembers are adjacent to some members of the alliance. The concept of alliances was introduced by Kristiansen et al. (2004). After that many results concerning the complexity for finding a minimum alliance were shown. It was shown that the problem of finding a minimum alliance of any variant is NP-hard in general. For some variants, it was shown that the problem becomes polynomial-time solvable when restricted to trees or series–parallel graphs. In this paper, we show that the problems for all variants are efficiently solvable for much larger graph classes. We present a polynomial-time algorithm for graphs of bounded clique-width. We also show that the problem is fixed-parameter tractable when parameterized by the vertex cover number.

Many real world operational research problems, such as frequency assignment and exam timetabling, can be reformulated as graph colouring problems (GCPs). Most algorithms for the GCP operate under the assumption that its constraints are fixed, allowing us to model the problem using a static graph. However, in many real-world cases this does not hold and it is more appropriate to model problems with constraints that change over time using an edge dynamic graph. Although exploring methods for colouring dynamic graphs has been identified as an area of interest with many real-world applications, to date, very little literature exists regarding such methods. In this paper we present several heuristic methods for modifying a feasible colouring at time-step t into an initial, but not necessarily feasible, colouring for a “similar” graph at time-step t+1t+1 . We will discuss two cases; (1) where changes occur at random, and (2) where probabilistic information about future changes is provided. Experimental results are also presented and the benefits of applying these particular modification methods are investigated.

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of $n$ vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex $u$ to vertex $v$ is a path from $u$ to $v$ where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a $(u,v)$-journey for any pair of vertices $u,v,~u\not= v$. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can \emph{freely choose} availability instances for all edges and aims for temporal connectivity with very small \emph{cost}; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in $n$. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead \emph{given} a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless $P=NP$. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be \emph{minimal}, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least $n \log{n}$ labels.

Constructing a graph to represent the structure among data objects plays a fundamental role in various data mining tasks with graph-based learning. Since traditional pairwise distance-based graph construction is sensitive to noise and outliers, sparse representation based graphs (e.g., ź1-graphs) have been proposed in the literature. Although ź1-graphs prove powerful and robust for many graph-based learning tasks, it suffers from weak locality and high computation costs. In this paper, we propose a locality weighted sparse representation (LWSR), which aims for good preservation of the locality structure among data objects and a significant reduction of the computation time. LWSR approximates each object as a sparse linear combination of its nearest neighbors, and weights their corresponding coefficients by their distances to the target object. Experimental results show that LWSR-graph based learning methods outperform state-of-the-art methods in both effectiveness and efficiency for graph-based learning.

Graph is widely used to model data in various applications. With the rapid growth of many emerging applications such as Internet of Things, it is urgent to require the processing capability on large scale graphs with billions of vertices. Web graph is a typical case of graph data that is widely used for analyzing the structure, behavior and evolution of the World Wide Web. In this paper, we focus on optimal representation of large-scale Web graphs. Our work is motivated by the need of fit large-scale graphs into the main memory and carry out analyze on them. By analyzing the adjacency matrix of Web graphs, we find two characteristics on the distribution of 1s in the matrix. Firstly, only a very small proportion of elements in the matrix are 1s. Secondly, majority of 1s gather around the principal diagonal and form a few number of clusters in the matrix. Based on these characteristics, we first develop a clustering mechanism to locate the clusters of 1s in the adjacency matrix. Then, we combine this clustering mechanism with a structure named K2-tree and propose an approach for representing large-scale Web graphs compactly. Basic idea of the approach is trying to compress a large number of zeros as a single zero. Experimental results show that, our approach not only reduces the space for representing a Web graph, but also reduces the time consumption for operations such as retrieving neighbors of any nodes on the graph; compared with existing approaches, our approach achieves the best space/time tradeoff.