🤖 AI Summary
This work addresses efficient algorithms for bipartiteness-related graph problems—including $K_t$-subgraph covering, odd-cycle transversal, and maximum weighted cut—by introducing a novel graph width parameter: *bipartite treewidth* (btw), which unifies treewidth and odd-cycle transversal number. Building upon an enhanced tree decomposition framework, we develop the first dynamic programming paradigm tailored to btw, yielding fixed-parameter tractable (FPT) algorithms for multiple NP-hard problems. Furthermore, for any 2-connected graph $H$, we establish a complexity dichotomy theorem for $H$-odd-subgraph packing and related problems: under the btw parameterization, we precisely delineate the boundary between para-NP-completeness and XP-solvability. Integrating parameterized algorithmics, dynamic programming, and structural graph modeling, this work substantially extends the applicability of the treewidth framework to non-bipartite graph problems.
📝 Abstract
We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that $K_t$-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by bipartite treewidth. We provide the following complexity dichotomy when $H$ is a 2-connected graph, for each of $H$-Subgraph-Packing, $H$-Induced-Packing, $H$-Scattered-Packing, and $H$-Odd-Minor-Packing problem: if $H$ is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if $H$ is non-bipartite, then it is solvable in XP-time. We define 1-${cal H}$-treewidth by replacing the bipartite graph class by any class ${cal H}$. Most of the technology developed here works for this more general parameter.
