Global cycle properties in locally isometric graphs

TL;DR

This paper investigates the cycle properties of locally isometric graphs, proving NP-completeness of Hamilton cycle detection for certain degrees and characterizing cycle extendability in low-degree cases, confirming a conjecture for a specific subclass.

Contribution

It provides structural characterizations of locally isometric graphs with maximum degree at most 6 and proves Ryjacek's conjecture for a subclass of locally connected graphs.

Findings

Hamilton cycle problem is NP-complete for locally isometric graphs with degree ≤ 8.

Locally isometric graphs with degree ≤ 6 are fully cycle extendable.

Such graphs are weakly pancyclic.

Abstract

A graph G is locally isometric if the subgraph induced by the neighbourhood of every vertex is an isometric subgraph of G. It is shown that the hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is NP-complete. Structural characterizations of locally isometric graphs, with maximum degree at most 6, that are fully cycle extendable, are established and these results are used to show that locally isometric graphs with maximum degree at most 6 are weakly pancyclic. This proves Ryjacek's conjecture for a subclass of locally connected graphs.