The (theta, wheel)-free graphs Part IV: Induced paths and cycles

Abstract

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class.

Induced Disjoint Paths in AT-free graphs

“…For claw-free graphs, Induced Disjoint Paths stays NP-complete as well. This is shown by Fiala et al [13] even for line graphs, which form a subclass of claw-free graphs, and very recently by Radovanović, Trotignon and Vušković [38] even for line graphs of triangle-free chordless graphs. In addition, Fiala et al [13] showed that Induced Disjoint Paths can be solved in polynomial time for claw-free graphs if k is fixed.…”

Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs

“…Only results for Induced Disjoint Paths are known and these hold for a slightly more general problem definition (see Section 6). Namely, Induced Disjoint Paths is linear-time solvable for circular-arc graphs [10]; polynomial-time solvable for chordal graphs [1], AT-free graphs [11], graph classes of bounded mim-width [13]; and NP-complete for claw-free graphs [6], line graphs of triangle-free chordless graphs [24] and thus for (theta,wheel)-free graphs, and for planar graphs; the last result follows from a result of Lynch [21] (see [11]).…”

“…Moreover, Induced Disjoint Paths is XP with parameter k for (theta,wheel)free graphs [24] and even FPT with parameter k for claw-free graphs [9] and planar graphs [15]; the latter can be extended to graph classes of bounded genus [18].…”

The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs

“…In each case, this theorem fully describes the structure of the most general graph in the class, and could therefore be used to provide algorithms for several combinatorial optimisation problems. This is done in Parts III and IV of this series (see [27] and [28]), where polynomial-time algorithms for finding maximum weighted clique and stable set, for optimal coloring and for induced version of k-linkage problem (for k fixed) are obtained for the class of (theta,wheel)-free graphs. We note that among the 16 classes described in Table 1, only universally signable graphs (line 0 from the table) have a (previously known) decomposition theorem.…”