On game chromatic vertex-critical graphs

TL;DR

This paper investigates the concept of vertex-criticality in graphs based on various game chromatic numbers, providing characterizations and exploring differences in how these invariants change when vertices are removed.

Contribution

It introduces and characterizes different types of game-vertex-critical graphs for multiple game chromatic invariants, highlighting their properties and connectivity conditions.

Findings

The difference in game chromatic numbers after vertex removal can be arbitrarily large.

Characterizations of 2- and 3-vertex-critical graphs are provided.

Some critical graphs are not necessarily connected, except for certain invariants.

Abstract

Several games that arise from graph coloring have been introduced and studied. Let $\varphi$ denote a graph invariant that arises from such a game. If $G$ is a graph and $\varphi(G-x)\neq \varphi(G)=k$, $k \geq 1$, holds true for every vertex $x \in V(G)$, then $G$ is called a $k$-$\varphi$-game-vertex-critical graph. We study the concept of $\varphi$-game-vertex-criticality for $\varphi \in \{\chi_g, \chi_i, \chi_{ig}^{A}, \chi_{ig}^{AB}\}$, where $\chi_g$ denotes the standard game chromatic number, $\chi_i$ denotes the indicated game chromatic number and $\chi_{ig}^{A}$, $\chi_{ig}^{AB}$ denote two versions of the independence game chromatic number. Since the game chromatic number $\varphi(G-x)$ can either decrease or increase with respect to $\varphi(G)$, we distinguish between lower, upper and mixed vertex-criticality. We show that for $\varphi \in \{\chi_g, \chi_{ig}^{A}, \chi_{ig}^{AB}\}$ the difference $\varphi(G)-\varphi(G-x)$, $x \in V(G)$, can be arbitrarily large. A characterization of $2$-$\varphi$-game-vertex-critical and (connected) $3$-$\varphi$-lower-game-vertex-critical graphs for all $\varphi \in \{\chi_g, \chi_i, \chi_{ig}^{A}, \chi_{ig}^{AB}\}$ is given. It is shown that $\chi_g$-game-vertex-critical, $\chi_{ig}^{A}$-game-vertex-critical and $\chi_{ig}^{AB}$-game-vertex-critical graphs are not necessarily connected. However, it is also shown that $\chi_i$-lower-game-vertex-critical graphs are always connected.