On the Structure of (Even Hole, Kite)-Free Graphs

Abstract

A hole is an induced cycle with at least four vertices. A hole is even if it has an even number of vertices. Even-hole-free graphs are being actively studied in the literature. It is not known whether even-hole-free graphs can be optimally colored in polynomial time. A diamond is the graph obtained from the clique of four vertices by removing one edge. A kite is a graph consists of a diamond and another vertex adjacent to a vertex of degree two in the diamond. Kloks et al. (J Combin Theory B 99:733–800, 2009) designed a polynomial-time algorithm to color an (even hole, diamond)-free graph. In this paper, we consider the class of (even hole, kite)-free graphs which generalizes (even hole, diamond)-free graphs. We prove that a connected (even hole, kite)-free graph is diamond-free, or the join of a clique and a diamond-free graph, or contains a clique cutset. This result, together with the result of Kloks et al., imply the existence of a polynomial time algorithm to color (even hole, kite)-free graphs. We also prove (even hole, kite)-free graphs are $$\beta $$ -perfect in the sense of Markossian, Gasparian and Reed. Finally, we establish the Vizing bound for (even hole, kite)-free graphs.