Combinatorial Problems on HH-graphs

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

Mim-Width III. Graph powers and generalized distance domination problems

We generalize the family of (σ,ρ) problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as Distance-r Dominating Set and Distance-r Independent Set. We show that these distance problems are in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mimwidth + solution size. We show that powers of graphs of tree-width w − 1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.publishedVersio

Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure

We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Independent Set and related problems. In the previous paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a min-max graph invariant related to tree decompositions. Bounded tree-independence number implies both $(\mathrm{tw},\omega)$-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Set problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In this paper, we consider six graph containment relations and for each of them characterize the graphs $H$ for which any graph excluding $H$ with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a $K_5$ minus an edge or the $4$-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most $3$ vertices, while excluding a complete bipartite graph $K_{2,q}$ implies the existence of a tree decomposition with independence number at most $2(q-1)$. Our constructive proofs are obtained using a variety of tools, including $\ell$-refined tree decompositions, SPQR trees, and potential maximal cliques. They imply polynomial-time algorithms for the Independent Set and related problems in an infinite family of graph classes; in particular, the results apply to the class of $1$-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.Comment: 46 pages; abstract has been shortened due to arXiv requirements. A previous arXiv post (arXiv:2111.04543) has been reorganized into two parts; this is the second of the two part

Treewidth versus clique number. II. Tree-independence number

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(\mathrm{tw},\omega)$-bounded. While $(\mathrm{tw},\omega)$-bounded graph classes are known to enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem whether $(\mathrm{tw},\omega)$-boundedness also has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by means of a new min-max graph invariant related to tree decompositions. We define the independence number of a tree decomposition $\mathcal{T}$ of a graph as the maximum independence number over all subgraphs of $G$ induced by some bag of $\mathcal{T}$. The tree-independence number of a graph $G$ is then defined as the minimum independence number over all tree decompositions of $G$. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes will be given in the third paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A previous version of this arXiv post has been reorganized into two parts; this is the first of the two parts (the second one is arXiv:2206.15092

On H-Topological Intersection Graphs

Biro, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. Our paper is the first study of the recognition and dominating set problems of this large collection of intersection classes of graphs.We negatively answer the question of Biro, Hujter, and Tuza who asked whether H-graphs can be recognized in polynomial time, for a fixed graph H. Namely, we show that recognizing H-graphs is NP-complete if H contains the diamond graph as a minor. On the other hand, for each tree T, we give a polynomial-time algorithm for recognizing T-graphs and an O(n(4))-time algorithm for recognizing K-1,K-d-graphs. For the dominating set problem (parameterized by the size of H), we give FPT- and XP-time algorithms on K-1, d-graphs and H-graphs, respectively. Our dominating set algorithm for H-graphs also provides XP-time algorithms for the independent set and independent dominating set problems on H-graphs