Structure of (bull, diamond)-free graphs and its applications

Abstract

This paper discusses the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a partitionable (bull, diamond)-free graph can be found in polynomial time. Additionally, we show that the cop number of a (bull, diamond)-free graph containing a triangle is at most two less than its diameter. Furthermore, the cop number of a connected ($P_n$, bull, diamond)-free graph with a triangle, is at most $n-3$, for any natural number $n>3$. We also discuss a couple of applications of the structural theorem of the (bull, diamond)-free graphs in the conclusions.