Graphs with span 1 and shortest optimal walks

TL;DR

This paper investigates the concept of span in graphs, analyzing the differences between vertex and edge spans, establishing bounds, characterizing graphs with minimal strong vertex span, and providing an algorithm for optimal traversal.

Contribution

It introduces bounds on vertex and edge spans, characterizes graphs with strong vertex span 1, and presents an algorithm for minimal traversal moves.

Findings

Vertex and edge span differ by at most 1.

Interval graphs have strong vertex span equal to 1.

An algorithm computes minimal moves for graph traversal.

Abstract

A span of a given graph $G$ is the maximum distance that two players can keep at all times while visiting all vertices (edges) of $G$ and moving according to certain rules, that produce different variants of span. We prove that the vertex and edge span of the same variant can differ by at most 1 and present a graph where the difference is exactly 1. For all variants of vertex span we present a lower bound in terms of the girth of the graph. Then we study graphs with the strong vertex span equal to 1. We present some nice properties of such graphs and show that interval graphs are contained in the class of graphs having the strong vertex span equal to 1. Finally, we present an algorithm that returns the minimum number of moves needed such that both players traverse all vertices of the given graph $G$ such that in each move the distance between players equals at least the chosen span of $G$.