A vertex and edge deletion game on graphs

TL;DR

This paper studies a combinatorial game involving vertex and edge deletions on graphs, extending known results, proving a conjecture, and revealing regularities in specific graph families.

Contribution

It extends the computation of nim-values to new graph families and proves a conjecture about graphs with one odd cycle.

Findings

Nim-values computed for new graph families

Conjecture on graphs with one odd cycle proved

Regularity observed in wheels and subgraphs

Abstract

Starting with a graph, two players take turns in either deleting an edge or deleting a vertex and all incident edges. The player removing the last vertex wins. We review the known results for this game and extend the computation of nim-values to new families of graphs. A conjecture of Khandhawit and Ye on the nim-values of graphs with one odd cycle is proved. We also see that, for wheels and their subgraphs, this game exhibits a surprising amount of unexplained regularity.