Dominating cycles and forbidden pairs containing a path of order 5

TL;DR

This paper characterizes certain forbidden pairs of subgraphs that guarantee the existence of dominating cycles in 2-connected graphs, specifically focusing on pairs involving a path of length 5.

Contribution

It proves that specific forbidden pairs ensure the longest cycle in 2-connected graphs is dominating, advancing understanding of graph cycle properties.

Findings

2-connected {P5, K4^-}-free graphs have a dominating longest cycle

2-connected {P5, W*}-free graphs have a dominating longest cycle

Identifies forbidden pairs guaranteeing dominating cycles

Abstract

A cycle is a graph is dominating if every edge of the graph is incident with a vertex of the cycle. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every $2$-connected $\{P_{5}, K_{4}^{-}\}$-free graph contains a longest cycle which is a dominating cycle. (ii) Every $2$-connected $\{P_{5}, W^{*}\}$-free graph contains a longest cycle which is a dominating cycle. Here $P_{5}$ is the path of order $5$, $K_{4}^{-}$ is the graph obtained from the complete graph of order $4$ by removing one edge, and $W^{*}$ is a graph obtained from two triangles and an edge by identifying one vertex in each.