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The $k$-CombDMR problem is that of determining whether an $n \times n$ distance matrix can be realised by $n$ vertices in some undirected graph with $n + k$ vertices. This problem has a simple solution in the case $k=0$. In this paper we show that this problem is polynomial time solvable for $k=1$ and $k=2$. Moreover, we provide algorithms to construct such graph realisations by solving appropriate 2-SAT instances. In the case where $k \geq 3$, this problem is NP-complete. We show this by a reduction of the $k$-colourability problem to the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time solvable problem of tree realisability for a given distance matrix., Comment: 27 pages, 5 figures

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an $n \times n$ matrix $D$ and a $\Delta \in \mathbb{N}$, the goal is to construct a $\Delta$-periodic temporal graph with $n$ vertices such that the duration of a fastest path from $v_i$ to $v_j$ is equal to $D_{i,j}$, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs have been well studied and understood since the 1960's (e.g. [Erd\H{o}s and Gallai, 1960], [Hakimi and Yau, 1965]). As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i.e. non-temporal) counterpart. First, we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the ``tree-likeness''. We prove a tight classification between such parameters that allow fixed-parameter tractability (FPT) and those which imply W[1]-hardness. We show that our problem is W[1]-hard when parameterized by the feedback vertex number (and therefore also any smaller parameter such as treewidth, degeneracy, and cliquewidth) of the underlying graph, while we show that it is in FPT when parameterized by the feedback edge number (and therefore also any larger parameter such as maximum leaf number) of the underlying graph., Comment: 57 pages, 10 figures

In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph $G$ and a Hamiltonian cycle $C_0$ of $G$, how can we compute a second Hamiltonian cycle $C_1 \neq C_0$ of $G$? Cedric Smith proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is an algorithm which computes the second Hamiltonian cycle in time $O(n \cdot 2^{(0.3-\varepsilon)n})$ time, for some positive constant $\varepsilon>0$, and in polynomial space, thus improving the state of the art running time for solving this problem. Our algorithm is based on a fundamental structural property of Thomason's lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a Hamiltonian graph $G$ with a given Hamiltonian cycle $C_0$ (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least $n - 4\alpha (\sqrt{n}+2\alpha)+8$, where $\alpha = \frac{\Delta-2}{\delta-2}$ and $\delta,\Delta$ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art., Comment: 28 pages, 4 algorithms, 5 figures

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have running time $O(n^4)$, where $n$ is the number of graph vertices. Exploiting the framework of parameterized complexity analysis, we explore possibilities for "linear-time FPT" algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time $O(2^{O(k)} + n +m)$ ($m$ being the number of graph edges) while at the same time, unless the SETH fails, there is no $2^{o(k)}n^2$-time algorithm.

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For $m$-edge and $n$-vertex graphs, it is well-known to be solvable in $O(m\sqrt{n})$ time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on (almost) linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings., Comment: Available as 'Online First' in Algorithmica

This work studies the parameterized complexity of finding secluded solutions to classical combinatorial optimization problems on graphs such as finding minimum s-t separators, feedback vertex sets, dominating sets, maximum independent sets, and vertex deletion problems for hereditary graph properties: Herein, one searches not only to minimize or maximize the size of the solution, but also to minimize the size of its neighborhood. This restriction has applications in secure routing and community detection., Comment: Compared to the previous version, this version additionally shows that Small Secluded s-t-Separator is fixed-parameter tractable parameterized by the combination of the solution size and the open neighborhood size (Theorem 3.5). To appear in Discrete Optimization

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GreedyMatching. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GreedyMatching is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GreedyMatching is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider two natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GreedyMatching on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is $\rho$, then for every $\epsilon>0$ the randomized MRG algorithm of [Aronson et al., Rand. Struct. Alg. 1995] gives a $(\rho-\epsilon)$-approximation for the maximum cardinality matching. We conjecture that a tight bound for $\rho$ is $\frac{2}{3}$; we prove our conjecture true for two subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for $\rho$) is a long-standing open problem [Poloczek and Szegedy; FOCS 2012]. This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem., Comment: 21 pages, 6 figures

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path, that is, given an undirected graph, find a maximum-length path in $G$. Longest Path is NP-hard in general but known to be solvable in $O(n^{4})$ time on $n$-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number $k$ to proper interval graphs, in $O(k^{9}n)$ time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value $k$ can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems., Comment: 34 pages, 1 figure, 1 algorithm, 4 reduction rules

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.

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval - one from the left and one from the right side. Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now, three decades after the introduction of tolerance graphs. In this article we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane. Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APX-hard on multitolerance graphs, solving thus a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes.

Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely \emph{simple-triangle} graphs. Simple-triangle graphs - also known as \emph{PI} graphs (for Point-Interval) - are the intersection graphs of triangles that are defined by a point on a line $L_{1}$ and an interval on a parallel line $L_{2}$. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between $L_{1}$ and $L_{2}$ and of trapezoids between $L_{1}$ and $L_{2}$, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simple-triangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simple-triangle graphs can be recognized in polynomial time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely the recognition of \emph{linear-interval orders}, i.e. of partial orders $P=P_{1}\cap P_{2}$, where $P_{1}$ is a linear order and $P_{2}$ is an interval order. This is one of the first results on recognizing partial orders $P$ that are the intersection of orders from two different classes $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. In complete contrast to this, partial orders $P$ which are the intersection of orders from the same class $\mathcal{P}$ have been extensively investigated, and in most cases the complexity status of these recognition problems has been already established., Comment: 27 pages, 4 figures, 5 algorithms

It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. The conjecture has been proved under some - rather strong - \emph{structural} assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Our main result in this article is that the above conjecture is true for every graph $G$ that admits a tolerance representation with exactly one unbounded vertex; note here that this assumption concerns only the given tolerance \emph{representation} $R$ of $G$, rather than any structural property of $G$. Moreover, our results imply as a corollary that the conjecture of Golumbic, Monma, and Trotter is true for every graph $G=(V,E)$ that has no three independent vertices $a,b,c\in V$ such that $N(a) \subset N(b) \subset N(c)$; this is satisfied in particular when $G$ is the complement of a triangle-free graph (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are constructive, in the sense that, given a tolerance representation $R$ of a graph $G$, we transform $R$ into a bounded tolerance representation $R^{\ast}$ of $G$. Furthermore, we conjecture that any \emph{minimal} tolerance graph $G$ that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic, Monma, and Trotter, it suffices to prove our conjecture., Comment: 58 pages, 9 figures. A preliminary conference version appeared in the Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC), Jeju Island, Korea, December 2010, Volume 1, pages 230-240

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete. Furthermore we present a subclass of graphs with diameter 2 that admits a polynomial algorithm for 3-coloring. For graphs with diameter 3 we establish the complexity of 3-coloring by proving that for every $\varepsilon \in [0,1)$, 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with $n$ vertices and minimum degree $\delta=\Theta(n^{\varepsilon})$. Moreover, assuming ETH, we provide three different amplifications of our hardness results to obtain for every $\varepsilon \in [0,1)$ subexponential lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree $\delta=\Theta(n^{\varepsilon})$. Finally, we provide a 3-coloring algorithm with running time $2^{O(\min\{\delta\Delta,\frac{n}{\delta}\log\delta\})}$ for graphs with diameter 3, where $\delta$ (resp. $\Delta $) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this algorithm is the first subexponential algorithm for graphs with $\delta=\omega(1)$ and for graphs with $\delta=O(1)$ and $\Delta=o(n)$. Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree if the input graph is $\delta=\Theta(n^{\varepsilon})$, where $\varepsilon \in [1/2,1)$., Comment: 25 pages, 10 figures, 2 tables, 1 algorithm

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications. Several efficient algorithms for optimization problems that are NP-hard on general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs - namely, the problem of deciding whether a given graph is a tolerance graph - as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf. the book on tolerance graphs \cite{GolTol04}) since their introduction in 1982 \cite{GoMo82}. In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them. For our reduction we extend the notion of an \emph{acyclic orientation} of permutation and trapezoid graphs. Our main tool is a new algorithm that uses \emph{vertex splitting} to transform a given trapezoid graph into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it has been proved a powerful tool also in the design of efficient recognition algorithms for other classes of graphs \cite{MC-Trapezoid}., Comment: 12 pages, 4 figures, 1 algorithm, Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), Nancy, France, March 2010