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We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

12th International Symposium on Parameterized and Exact Computation (IPEC 2017); International audience; Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γk. A graph class G has the SQGC property if every graph G ∈ G has treewidth O(bcg(G)c) for some 1 ≤ c < 2. The SQGC property is important for algo- rithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects.

It is widely accepted that necessity comes in different varieties, often called ‘kinds': metaphysical necessity, logical necessity, natural necessity, conceptual necessity, moral necessity, to name but a few – and the same goes for the varieties of possibility. What is usually not fully appreciated, however, is that modal variety is not simply ‘unidimensional': it does not only involve one main variable – kind, whose values are the particular kinds of necessity. Rather, I argue, it is ‘bidimensional', involving two distinct variables – domain and strength. In the first part of the paper, I introduce and develop the proposed bidimensional picture of modal variety, defending it against the common, unidimensional one. In the second part, I consider how the main available accounts of necessities and their relations rely, at least to a significant extent, on the latter picture, pointing out important limitations that they face as a result. I also show how, accordingly, alternative accounts based on a clear and systematic distinction between domain and strength would overcome those limitations. I conclude that, beyond the particular bidimensionalist view defended, our understanding of the modal realm may benefit from more direct debate on whether and how it is multidimensional.

Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp.866--893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E.D. Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems g graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work., Comment: An an earlier version of this paper appeared in SODA 2010. That paper contained preliminary versions of some of the results of this paper

The vertex cover problem and the dominating set problem are two well-known problems in graph theory. Their goal is to find the minimum size of a vertex subset satisfying some properties. Both hold a connected version, which imposes that the vertex subset must induce a connected component. To study the interdependence between the connected version and the original version of a problem, the Price of Connectivity ( $$PoC$$ ) was introduced by Cardinal and Levy (Theor Comput Sci 411(26–28):2581–2590, 2010) and Levy (Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Universite libre de Bruxelles, Brussels, 2009) as the ratio between invariants from the connected version and the original version of the problem. Camby et al. (Discret Math Theor Comput Sci 16:207–224, 2014) for the vertex cover problem, Camby and Schaudt (Discret Appl Math 177:53–59, 2014) for the dominating set problem characterized some classes of $$PoC$$ -Near-Perfect graphs, hereditary classes of graphs in which the Price of Connectivity is bounded by a fixed constant. Moreover, only for the vertex cover problem, Camby et al. (2014) introduced the notion of critical graphs, graphs that can appear in the list of forbidden induced subgraphs characterization. By definition, the Price of Connectivity of a critical graph is strictly greater than that of any proper induced subgraph. In this paper, we prove that for the vertex cover problem, every critical graph is either isomorphic to a cycle on 5 vertices or bipartite. To go further in the previous studies, we also present conjectures on $$PoC$$ -Near-Perfect graphs and critical graphs with the help of the computer software GraphsInGraphs (Camby and Caporossi in Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10, 2016). Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent.

International audience; For a fixed connected graph H, the {H}-M-Deletion problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f (tw) · n O(1) , where tw is the treewidth of G. We determine the asymptotically optimal function f (tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) = 2Θ(tw) if H is a contraction of the chair or the banner, and f(tw) = 2Θ(tw·log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2Θ(tw·log tw) ·nO(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F = {K5, K3,3}. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

International audience; In parameterized complexity, a {\em kernelization algorithm} can be seen as a reduction of a parameterized problem to itself, so that the produced equivalent instance has size depending exclusively on the parameter. If this size is polynomial, then then we say that the parameterized problem in question admits a polynomial kernelization algorithm. Kernelization can be seen as a formalization of the notion of preprocessingand has occupied a big part of the research on Multi-variate Algorithmics. The first algorithmic meta-theorem on kernelizationappeared in \href{https://arxiv.org/pdf/0904.0727}{\green{[{\sl H.~L. Bodlaender, F.~V. Fomin, D.~Lokshtanov, E.~Penninkx, S.~Saurabh, and D.~M. Thilikos. (Meta) kernelization. J. {ACM}, 63(5):44:1--44:69, 2016}\,]}} and unified a large family of previously known kernelization results on problems defined on topologically embeddable graphs.In this exposition we present the central results of this paper. During our presentation we pay attention to the abstractions on which the results where founded and take into account the subsequent advancements on this topic.

The relevance of the Casimir effect, discovered in 1948, has recently been pointed out in studies on materials such as graphene and high-temperature superconducting cuprates. In particular, the relationship between Casimir energy and the energy of a superconducting condensate with anisotropy characterized by high bidimensionality has already been discussed in certain theoretical scenarios. Using this proposal, this work describes the relationship between the effective mass of the charge carriers ( m ⁎ = α m e ) and the macroscopic parameters characteristic of several families of high- T c superconducting cuprates (Cu-HTSC) that have copper and oxygen superconducting planes (Cu-O). We have verified that an expression exists that correlates the effective mass, the London penetration length in the plane λ a b , the critical temperature T c and the distance d between the equivalent superconducting planes of Cu-HTSC. This study revealed that the intersection between the asymptotic behavior of α as a function of T c and the line describing the optimal value of α ≃ 2 ( m ⁎ ≃ 2 m e ) indicates that a nonadiabatic region exists, which implies a carrier-lattice interaction and where the critical temperature can have its highest value in Cu-HTSC.

Two of the most widely used approaches to obtain polynomial-time approximation schemes (PTASs) on planar graphs are the Lipton-Tarjan separator-based approach and Baker’s approach. In 2005, Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial-time approximation schemes (EPTASs) for several problems, including C onnected D ominating S et and F eedback V ertex S et . In this work, we unify the two strengthened approaches to combine the best of both worlds. We develop a framework allowing the design of EPTAS on classes of graphs with the subquadratic grid minor (SQGM) property. Roughly speaking, a class of graphs has the SQGM property if, for every graph G from the class, the fact that G contains no t × t grid as a minor guarantees that the treewidth of G is subquadratic in t . For example, the class of planar graphs and, more generally, classes of graphs excluding some fixed graph as a minor, have the SQGM property. At the heart of our framework is a decomposition lemma stating that for “most” bidimensional problems on a graph class G with the SQGM property, there is a polynomial-time algorithm that, given a graph G ϵ G as input and an ϵ > 0, outputs a vertex set X of size ϵ ċ OPT such that the treewidth of G - X is f (ϵ). Here, OPT is the objective function value of the problem in question and f is a function depending only on ϵ. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors nor contractions. To the best of our knowledge, for many of these problems—including C ycle P acking , F -P acking , F -D eletion , M ax L eaf S panning T ree , or P artial r -D ominating S et —no EPTASs, even on planar graphs, were previously known. We also prove novel excluded grid theorems in unit disk and map graphs without large cliques. Using these theorems, we show that these classes of graphs have the SQGM property. Based on the developed framework, we design EPTASs and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs.

In a graph $$G=(V,E)$$G=(V,E), a set $$D \subseteq V$$D⊆V is said to be a dominating set of G if for every vertex $$u\in V{\setminus }D$$u?V\D, there exists a vertex $$v\in D$$v?D such that $$uv\in E$$uv?E. A secure dominating set of the graph G is a dominating set D of G such that for every $$u\in V{\setminus }D$$u?V\D, there exists a vertex $$v\in D$$v?D such that $$uv\in E$$uv?E and $$(D{\setminus }\{v\})\cup \{u\}$$(D\{v})?{u} is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.

Let G be a graph of order n . A subset S of V ( G ) is a dominating set of G if every vertex in V ( G ) ⧹ S is adjacent to at least one vertex of S . The domination polynomial of G is the polynomial D ( G , x ) = ∑ i = γ ( G ) n d ( G , i ) x i , where d ( G , i ) is the number of dominating sets of G of size i , and γ ( G ) is the size of a smallest dominating set of G , called the domination number of G . We say two graphs G and H are dominating equivalent if D ( G , x ) = D ( H , x ) . A graph G is said to be dominating unique , or simply D -unique, if D ( H , x ) = D ( G , x ) implies that H ≅ G . The goal of this paper is to find a new approach to determine the dominating uniqueness of graphs. In this paper, we define a new graph polynomial, called star polynomial, and introduced an analogy notion of star uniqueness of graphs. As an application, if G is a graph without isolated vertices, we show that a graph G is star unique if and only if G ‾ ∨ K m is dominating unique for each m ⩾ 0 . As a by-product, the dominating uniqueness of many families of dense graphs is also determined.

We consider boolean linear programming formulations of the independent set, the vertex and the edge dominating set problems and prove their polynomial-time solvability for classes of graphs with (augmented) constraint matrices having bounded minors in the absolute value.

A total dominating set of an undirected graph G=(V,E) is a set SV of vertices such that each vertex of G is adjacent to at least one vertex of S. A secure total dominating set of G is a set DV of vertices such that D is a total dominating set of G and each vertex vVD is adjacent to at least one vertex uD with the property that the set (D{u}){v} is a total dominating set of G. The secure total domination number st(G) of G is the minimum cardinality of a secure total dominating set of G. We establish bounds on the secure total domination number. In particular, we show that every graph G with no isolated vertices satisfies st(G)2(G), where (G) is the independence number of G. Further, we study the problem of finding the secure total domination number. We show that the decision version of the problem is NP-complete for chordal bipartite graphs, planar bipartite graphs with arbitrary large girth and maximum degree 3, split graphs and graphs of separability at most 2. Finally, we show that the optimisation version of the problem can be approximated in polynomial time within a factor of cln|V| for some constant c>1 and cannot be approximated in polynomial time within a factor of cln|V| for some constant c

A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. Following analogous studies in the literature related to independent sets, dominating sets, and total dominating sets, we study in this paper the class of graphs in which the connected dominating sets can be separated from the other vertex subsets by a linear weight function. More precisely, we say that a graph is connected-domishold if it admits non-negative real weights associated to its vertices such that a set of vertices is a connected dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We characterize the graphs in this non-hereditary class in terms of a property of the set of minimal cutsets of the graph. We give several characterizations for the hereditary case, that is, when each connected induced subgraph is required to be connected-domishold. The characterization by forbidden induced subgraphs implies that the class properly generalizes two well known classes of chordal graphs, the block graphs and the trivially perfect graphs. Finally, we study certain algorithmic aspects of connected-domishold graphs. Building on connections with minimal cutsets and properties of the derived hypergraphs and Boolean functions, we show that our approach leads to new polynomially solvable cases of the weighted connected dominating set problem., 32 pages, 8 figures

A set D of vertices in a graph G = ( V , E ) is called a 2-point set dominating set of G if for every set T ⊆ V − D there exists a non-empty set S ⊆ D containing at most two vertices such that the induced subgraph 〈 S ∪ T 〉 is connected. A set D ⊆ V ( G ) is called a global 2-point set dominating set of G if D is a 2-point set dominating set of both G and G ‾ . The global 2-point set domination number (2-point set domination number) is the minimum cardinality of a global 2-point set dominating set (2-point set dominating set) in G. In this paper we determine bounds on the global 2-point set domination number of a graph in terms of other graph invariants. We have also given relation between global 2-point set domination number and 2-point set domination number for some classes of graphs.

The outer-connected domination problem is abbreviated as the OCD problem.The OCD problem was known to be linear-time solvable on proper interval graphs.We solve the OCD problem on interval graphs in linear-time.We do so by developing a chainless decomposition of graphs. An outer-connected dominating set in a graph G = ( V , E ) is a set S ? V such that each vertex not in S is adjacent to at least one vertex in S and the subgraph induced by V ? S is connected. In Keil and Pradhan (2013) 6, Keil and Pradhan gave a linear-time algorithm for finding a minimum outer-connected dominating set in a proper interval graph. In this paper, we generalize their result to find a minimum outer-connected dominating set in an interval graph in linear time.

Let G = ( V , E ) be a connected graph, and k a positive integer. A subset D ⊆ V is a k-hop connected dominating set (k-CDS) if the subgraph of G induced by D is connected and, for every vertex v in G, there is a vertex u in D such that the distance between v and u is at most k. We study the problem of finding a minimum k-hop connected dominating set, denoted by the acronym Min k-CDS. Firstly, we prove that Min k-CDS is NP -hard on planar bipartite graphs of maximum degree 4 and on planar biconnected graphs of maximum degree 5. We present an inapproximability threshold for Min k-CDS on bipartite and on (1, 2)-split graphs, and we also prove that Min k-CDS is APX -hard on bipartite graphs of maximum degree 4. These results are shown to hold for every positive integer k. For k = 1 , the classical minimum connected dominating set problem, we present an integer linear programming formulation and show some classes of inequalities that define facets of the corresponding polytope. We also present an approximation algorithm for this case.

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X/{v}) ᑌ {u} is again a dominating set of G, in which case v is called a defender. The secure domination number of G is the cardinality of a smallest secure dominating set of G. In this paper, we show that every graph of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3.

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS o) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS o performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS o that is suitable for solving the minimum weight dominating set problem. The application of RLS o in graph mining is also briefly demonstrated.

A dominating set $$D \subseteq V$$D⊆V in a graph $$G=(V,E)$$G=(V,E) is a subset of vertices such that every vertex in V belongs to D or has at least one neighbour in D. In this paper we deal with the problem of finding an approximation of the dominating set of minimum size, i.e., the approximation of the minimum dominating set problem (MDS) in a distributed setting. A distributed algorithm that runs in a constant number of rounds, independent of the size of the network, is called local. In research on distributed local algorithms it is commonly assumed that each vertex has an unique identifier. However, as was shown by Goos et al., for certain classes of graphs (for example, lift-closed bounded degree graphs) identifiers are unnecessary and only a port numbering is needed. We confirm that the same remains true for the MDS up to a constant factor in the class of planar graphs. Namely, we present a local deterministic 694-approximation algorithm for the MDS in planar graphs in a model with a port numbering only. Moreover, our algorithm uses only short messages, i.e., in each round each node can send only a $$O(\log {|V|})$$O(log|V|)-bit message to each of its neighbours.

We solve the Eternal Dominating Set problem for proper interval graphs.We show that, in this case, the optimum equals the largest size of an independent set.For such graphs, there is no advantage in allowing multiple guards at a vertex. In this paper, we solve the Eternal Dominating Set problem for proper interval graphs. We prove that, in this case, the optimal value of the problem equals the largest size of an independent set. As a consequence, we show that the problem can be solved in linear time for such graphs. To obtain the result, we first consider another problem in which a vertex can be occupied by an arbitrary number of guards. We then derive a lower bound on the optimal value of this latter problem, and prove that, for proper interval graphs, it is the same as the optimum of the first problem.

Dominating sets play a vital role in day-to-day life problems. For-providing effective services in a location, central points are to be identified. This can easily be achieved by graph theoretic techniques. Such graphs and relevant parameters are introduced and extensively studied. One such parameter is complementary nil vertex edge dominating set(cnved-set). A vertex edge dominating set(ved-set) of a connected graph G with vertex set V is said to be a complementary nil vertex edge dominating set(cnved-Set) of G if and only if V - D is not a ved-set of G. A cnved-set of minimum cardinality is called a minimum cnved-set(mcnved-set)of G and this minimum cardinality is called the complementary nil vertex-edge domination number of G and is denoted by &#947cnve(G). We have given a characterization result for a ved-set to be a cnved-set and also bounds for this parameter are obtained.

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for $r$-Dominating Set and $r$-Scattered Set on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in \mathcal{F} are connected., 32 pages

An independent dominating set of a graph G is a subset D of V such that every vertex not in D is adjacent to at least one vertex of D and no two vertices in D are adjacent. The independent dominating set (IDS) problem asks for an independent dominating set with minimum cardinality. First, we show that the independent dominating set problem and the dominating set problem on cubic bipartite graphs are both NP-complete. As an additional result, we give an alternative and more direct proof for the NP-completeness of both the independent dominating set problem and the dominating set problem on at-most-cubic grid graphs. Next, we show that there are fixed-parameter tractable algorithms for the independent dominating set problem and the dominating set problem on at-most-cubic graphs, which run in O ( 3.3028 k + n ) and O ( 4.2361 k + n ) time, respectively. Moreover, we consider the weighted independent dominating set problem on ( k , ? ) -graphs. We show that the problem on ( 2 , 1 ) -graphs is NP-complete. We also show that the problem can be solved in linear time for ( 1 , 1 ) -graphs and in polynomial time for ( 1 , ? ) -graphs for constant ?, respectively.