Best and worst case permutations for random online domination of the path

TL;DR

This paper analyzes a randomized online algorithm for dominating path graphs, determining expected sizes, worst and best case performances, and characterizing permutations that lead to extremal outcomes.

Contribution

It provides a detailed analysis of the algorithm's expected performance and characterizes permutations for optimal and worst-case domination on paths.

Findings

Expected size of dominating set for path graphs

Characterization of permutations with worst and best performance

Refined analysis of extremal cases

Abstract

We study a randomized algorithm for graph domination, by which, according to a uniformly chosen permutation, vertices are revealed and added to the dominating set if not already dominated. We determine the expected size of the dominating set produced by the algorithm for the path graph $P_n$ and use this to derive the expected size for some related families of graphs. We then provide a much-refined analysis of the worst and best cases of this algorithm on $P_n$ and enumerate the permutations for which the algorithm has the worst-possible performance and best-possible performance. The case of dominating the path graph has connections to previous work of Bouwer and Star, and of Gessel on greedily coloring the path.