The detour covering number and cummerbund covering number of a graph

TL;DR

This paper introduces new graph concepts related to detours and cummerbunds, analyzing their properties and conditions for coverage, with results on minimum covering numbers under various graph constraints.

Contribution

It defines the detour and cummerbund covering numbers and explores their properties, providing bounds and extremal graphs under connectivity and girth conditions.

Findings

Conditions for graphs to be detour or cummerbund covered.

Bounds on minimum covering numbers based on graph properties.

Characterization of extremal graphs for 2-connected bipartite graphs.

Abstract

We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie in a detour. A graph is said to be detour covered if every vertex lies in a detour. The cummerbund covering number and cummerbund covered graphs are defined similarly. Some of the main results are as follows. (1) Minimum degree and forbidden subgraph conditions that ensure a graph to be cummerbund covered or detour covered. (2) The minimum cummerbund covering number and minimum detour covering number of a graph with connectivity or girth conditions. (3) The minimum cummerbund covering number of a $2$-connected bipartite graph and the extremal graphs.