Dominated balanced separators in wheel-induced-minor-free graphs

TL;DR

This paper proves that graphs excluding a fixed wheel as an induced minor have balanced separators dominated by a bounded number of vertices, confirming a conjecture for this class of graphs.

Contribution

It establishes the conjecture for wheel-induced-minor-free graphs, expanding understanding of graph separators in this class.

Findings

Confirmed the conjecture for wheel-induced-minor-free graphs

Established existence of bounded dominated balanced separators in these graphs

Extended previous results to a new class of minor-free graphs

Abstract

Gartland and Lokshtanov conjectured that every graph that excludes some planar graph as an induced minor has a balanced separator, that is, a separator whose deletion leaves every component with no more than half of the vertices of the graph, which is dominated by a bounded number of vertices. We confirm this conjecture for excluding any fixed wheel, that is, a cycle together with a universal vertex, as an induced minor.