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A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-_free_ if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $$c$$ for which every (theta, triangle)-free graph $$G$$ has treewidth at most $$c\log (|V(G)|)$$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $$G$$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $$k$$-Coloring (for fixed $$k$$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.
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Induced Subgraphs and Tree Decompositions IV. (Even Hole, Diamond, Pyramid)-Free Graphs
A hole in a graph $$G$$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $$K_4$$ by removing an edge. A pyramid is a graph consisting of a vertex $$a$$ called the apex and a triangle $$\{b_1, b_2, b_3\}$$ called the base, and three paths $$P_i$$ from $$a$$ to $$b_i$$ for $$1 \leq i \leq 3$$, all of length at least one, such that for $$i \neq j$$, the only edge between $$P_i \setminus \{a\}$$ and $$P_j \setminus \{a\}$$ is $$b_ib_j$$, and at most one of $$P_1$$, $$P_2$$, and $$P_3$$ has length exactly one. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-free if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, pyramid, $$K_4$$)-free graphs of arbitrarily large treewidth. Here, we show that for every $$t$$, (even hole, pyramid, diamond, $$K_t$$)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses “non-crossing decompositions” methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of “non-crossing decompositions” to prove bounded treewidth in a graph class of unbounded maximum degree.
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- Award ID(s):
- 2120644
- PAR ID:
- 10516145
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 30
- Issue:
- 2
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/11623
