2,992 research outputs found

    Recovering sparse graphs

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    We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d,k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in classes of sparse graphs. We derive as a corollary that if G_0 is a graph class with bounded expansion, then the first order model checking is fixed parameter tractable in the class of all graphs that can obtained from a graph G from G_0 by complementing on at most k arbitrary subsets of the vertex set of G; this implies an earlier result that the first order model checking is fixed parameter tractable in graph classes interpretable in classes of graphs with bounded maximum degree

    Combinatorial Problems on HH-graphs

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    Bir\'{o}, Hujter, and Tuza introduced the concept of HH-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph HH. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on HH-graphs. We show that for any fixed HH containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on HH-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on HH-graphs. Namely, when HH is a cactus the clique problem can be solved in polynomial time. Also, when a graph GG has a Helly HH-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the kk-clique and list kk-coloring problems are FPT on HH-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number

    Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

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    Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and P_4P\_4-sparseness). We believe that our most important result is an O(k4â‹…n+m){\cal O}(k^4 \cdot n + m)-time algorithm for computing a maximum matching where kk is either the modular-width or the P_4P\_4-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, P_4P\_4-lite graphs, P_4P\_4-extendible graphs and P_4P\_4-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width

    Partitioning Perfect Graphs into Stars

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    The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor

    Colouring exact distance graphs of chordal graphs

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    For a graph G=(V,E)G=(V,E) and positive integer pp, the exact distance-pp graph G[♮p]G^{[\natural p]} is the graph with vertex set VV and with an edge between vertices xx and yy if and only if xx and yy have distance pp. Recently, there has been an effort to obtain bounds on the chromatic number χ(G[♮p])\chi(G^{[\natural p]}) of exact distance-pp graphs for GG from certain classes of graphs. In particular, if a graph GG has tree-width tt, it has been shown that χ(G[♮p])∈O(pt−1)\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1}) for odd pp, and χ(G[♮p])∈O(ptΔ(G))\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G)) for even pp. We show that if GG is chordal and has tree-width tt, then χ(G[♮p])∈O(p t2)\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2) for odd pp, and χ(G[♮p])∈O(p t2Δ(G))\chi(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 \Delta(G)) for even pp. If we could show that for every graph HH of tree-width tt there is a chordal graph GG of tree-width tt which contains HH as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width tt. While we cannot do this, we show that for every graph HH of genus gg there is a graph GG which is a triangulation of genus gg and contains HH as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which arise from reviewers' comment
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