A Path Variant of the Explorer Director Game on Graphs

TL;DR

This paper investigates a variant of the Explorer-Director game on graphs, analyzing how the maximum number of visited vertices differs when players choose distance versus path length, revealing unbounded differences.

Contribution

It introduces a path-based variant of the game and proves that the difference in visited vertices between the two variants can be arbitrarily large.

Findings

The difference between $f_p(G,v)$ and $f_d(G,v)$ can be arbitrarily large.

Constructed graphs demonstrate unbounded gaps in game outcomes.

Theoretical bounds on the disparity between the two game variants.

Abstract

The Explorer-Director game, first introduced by Nedev and Muthukrishnan (2008), simulates a Mobile Agent exploring a ring network with an inconsistent global sense of direction. Two players, the Explorer and the Director, jointly control a token's movement on the vertices of a graph $G$ with initial location $v$. Each turn, the Explorer calls any valid distance, $d$, aiming to maximize the number of vertices the token visits, and the Director moves the token to any vertex distance $d$ away aiming to minimize the number of visited vertices. The game ends when no new vertices can be visited, assuming optimal play, and we denote the total number of visited vertices by $f_d(G,v)$. Here we study a variant where, if the token is on vertex $u$, the Explorer is allowed to select any valid \emph{path length}, $\ell$, and the Director now moves the token to any vertex $v$ such that $G$ contains a $uv$ path of length $\ell$. The corresponding parameter is $f_p(G,v)$. In this paper, we explore how far apart $f_d(G,v)$ and $f_p(G,v)$ can be, proving that for any $n$ there are graphs $G$ and $H$ with $f_p(G,v)-f_d(G,v)>n$ and $f_d(H,v)-f_p(H,v)>n$.