Games for width parameters and monotonicity

TL;DR

This paper introduces a unifying search game framework for various graph and matroid width parameters, providing new characterizations, monotonicity results, and simplified proofs for known properties.

Contribution

It develops a general framework for width parameters via search games, characterizes monotone variants, and offers a shorter proof relating matroid and graph tree-width.

Findings

Monotone search games coincide with certain tree decompositions.

Determining the winner in a class of scenarios is in NP.

Matroid tree-width of a graphic matroid is bounded by the graph's tree-width.

Abstract

We introduce a search game for two players played on a "scenario" consisting of a ground set together with a collection of feasible partitions. This general setting allows us to obtain new characterisations of many width parameters such as rank-width and carving-width of graphs, matroid tree-width and GF(4)-rank-width. We show that the monotone game variant corresponds to a tree decomposition of the ground set along feasible partitions. Our framework also captures many other decompositions into "simple" subsets of the ground set, such as decompositions into planar subgraphs. Within our general framework, we take a step towards characterising monotone search games. We exhibit a large class of "monotone" scenarios, i.e. of scenarios where the game and its monotone variant coincide. As a consequence, determining the winner is in NP for these games. This result implies monotonicity for all our search games, that are equivalent to branch-width of a submodular function. Finally, we include a proof showing that the matroid tree-width of a graphic matroid is not larger than the tree-width of the corresponding graph. This proof is considerably shorter than the original proof and it is purely graph theoretic.