A structure theorem for graphs with no cycle with a unique chord and its consequences

Abstract

We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraphs of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are in C.This has several consequences: an O(nm)-time algorithm to decide whether a graph is in C, an O(n + m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.

Complexity of colouring problems restricted to unichord-free and { square,unichord }-free graphs

“…}-free, ∆ ≥ 4 {sq.,un. }-free, ∆ = 3 vertex-colouring Polynomial [12] Polynomial [12] Polynomial [12] edge-colouring NP-complete [8] Polynomial [8] NP-complete [8] total-colouring NP-complete [5] Polynomial [5] Polynomial * Table 1: Computational complexity of colouring problems restricted to unichord-free and to {square,unichord}-free graphs -star indicates result established in the present paper.…”

“…In the present work, we consider total-colouring restricted to {square,unichord}-free graphs -that is, graphs that do not contain (as an induced subgraph) a cycle with a unique chord nor a square. The class of unichord-free graphs was studied by Trotignon and Vušković [12]. They give a structure theorem for the class, and use it to develop algorithms for recognition and vertex-colouring.…”

“…Basically, this structure result states that every unichord-free graph can be built starting from a restricted set of basic graphs and applying a series of known "gluing" operations. The following results are obtained in [12] for unichord-free graphs: an O(nm) recognition algorithm, an O(nm) algorithm for optimal vertex-colouring, an O(n + m) algorithm for maximum clique, and the NPcompleteness of the maximum stable set problem.…”

“…Machado, Figueiredo and Vušković [8] investigated whether the structure results of [12] could be applied to obtain a polynomial-time algorithm for the edge-colouring problem restricted to unichord-free graphs. The authors obtained a negative answer by establishing the NP-completeness of the edge-colouring problem restricted to unichord-free graphs.…”

Even-hole-free graphs: A survey

“…Interestingly, this is the first example in which a decomposition based method performs faster than the direct method (when both methods yield algorithms). We note that there are examples of recognition problems for which algorithms exist using only one of the methods, for example recognizing 3P C(∆, ·)-free graphs is only known by the direct method [7] and recognizing graphs with no cycle with a unique chord is only known by the decomposition method [45].…”