On the structure of (dart, odd hole)-free graphs
Chính T. Hoàng
TL;DR
This work analyzes (dart, odd hole)-free graphs and establishes a three-way structural dichotomy: such a graph is either perfect, has $\alpha(G) \le 2$, or admits a join/co-join partition into two smaller graphs. Leveraging this structure, the authors design a polynomial-time algorithm for optimal graph coloring by combining perfect-graph coloring, maximum-matchings in the complement, and recursive composition in the partition case, and show that these graphs are perfectly divisible with $\chi(G) \le \omega(G)^2$. They also prove that while coloring and large-stable-set computation are tractable, computing the maximum clique and minimum clique cover remain NP-hard for this class. The results advance understanding of the complexity landscape for odd-hole-free and related graph classes and provide practical coloring algorithms for this family, along with open directions such as identifying odd-hole-free graphs that are not perfectly divisible.
Abstract
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible.