Graphs with span 1 and shortest optimal walks

TL;DR

The paper studies graph spans that govern the maximal safety distance two players can maintain while visiting all vertices or edges under various movement rules. It formalizes these spans via weak homomorphisms, proves tight relations between edge and vertex variants, and derives girth-based lower bounds for vertex spans, including a sharp characterization of graphs with strong vertex span 1, notably showing interval graphs (and certain augmentations) fall into this class. It also provides constructions of infinite families with $\svSpan{G}=1$ and presents an algorithmic framework to compute the minimum number of steps in optimal walks using a graph-product-based approach, albeit with exponential-time complexity for the optimization step. Together, these results deepen understanding of span variants, their structural implications, and the algorithmic aspects of optimal walks in graphs.”, // all math wrapped appropriately in JSON; math terms are kept inline with LaTeX-style delimiters as required in the prompt

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 $G$.

Figures (3)

• Figure 1: A graph $G$ with $\sigma^{\boxtimes}_V(G)=\sigma^{\boxtimes}_E(G)+1$
• Figure 2: Non-chordal graph $G$ with $\sigma^{\boxtimes}_V(G)=1$
• Figure 3: An augmentation $G$ of an interval graph with respect to a non-minimal separating clique and a trivial graph with $\sigma^{\boxtimes}_V(G)=2$

Theorems & Definitions (33)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Remark 1.6
• Theorem 2.1
• proof
• Theorem 2.2
• proof