Dynamic programming on bipartite tree decompositions

TL;DR

This paper introduces bipartite treewidth as a graph parameter, explores its properties, and develops dynamic programming algorithms to solve various problems efficiently on graphs with small bipartite treewidth.

Contribution

It defines bipartite treewidth, analyzes its complexity, and provides fixed-parameter algorithms for key graph problems based on this parameter.

Findings

Bipartite treewidth generalizes treewidth and odd cycle transversal.

Certain problems are fixed-parameter tractable when parameterized by bipartite treewidth.

A dichotomy is established for problems based on the bipartiteness of the graph H.

Abstract

We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one "bipartite" vertex, while the width of such decomposition measures the number of "non-bipartite" vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that $K_t$-Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by btw. We also provide the following dichotomy when $H$ is a 2-connected graph: if $H$ is bipartite, then $H$-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing is para-NP-complete parameterized by btw while, if $H$ is non-bipartite, then the problem is solvable in XP-time.