Partisan Combinatorial Game of Edge and Vertex Removal on Graphs

TL;DR

This paper explores three variants of a partisan combinatorial game on graphs involving vertex and edge removal, analyzing strategic advantages and the impact of move restrictions.

Contribution

It introduces and analyzes three variants of a partisan graph game, highlighting the strategic implications of move privileges and restrictions.

Findings

Removing vertices offers a greater advantage than removing edges.

The variant where moves are mutually valid shows vertex removal as more beneficial.

The game variants reveal how move privileges influence game outcomes.

Abstract

We consider three variants of a partisan combinatorial game between two players, Left and Right, played on an undirected simple graph. Left is able to delete vertices (and incident edges) while Right is able to delete edges. This natural extension of a similar impartial game gives a clear advantage to one player by allowing them the ability to play on a small subgraph which the other player can not. Our last variant removes this advantage by assuming a move is valid for one player if and only if the other player has a valid move on the same graph. In this case, we show that the ability to remove a vertex is more advantageous compared to removing edges.