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.

翻译：我们重新审视一个图宽度参数——二分图树宽（bipartite treewidth），以及其对应的图分解——二分图树分解（bipartite tree decomposition）。二分图树宽可视为树宽与奇环横贯数（odd cycle transversal number）的共同推广。直观上，二分图树分解是一种树分解，其每个袋子（bag）诱导出近似二分图，且各粘附（adhesion）至多包含来自其他袋子中二分图部分的一个顶点；而此类分解的宽度衡量了袋子偏离二分图的程度。该分解源自Demaine、Hajiaghayi和Kawarabayashi [SODA 2010] 定义的树分解，并由Tazari [Th. Comp. Sci. 2012] 明确阐述。二分图树宽在解决与奇子式（odd-minors）相关的问题中似乎扮演关键角色，这些问题近期备受关注。作为构建高效求解此类问题理论的初步步骤，本文的主要目标是开发动态规划技术，以解决二分图树宽较小的图上的问题。对于此类图，我们提供了一系列para-NP完全性结果、FPT算法、XP算法，以及若干开放问题。特别地，我们证明 $K_t$-子图覆盖（$K_t$-Subgraph-Cover）、加权顶点覆盖/独立集（Weighted Vertex Cover/Independent Set）、奇环横贯（Odd Cycle Transversal）和最大加权割（Maximum Weighted Cut）问题在参数为二分图树宽时属于FPT。针对 $H$-子图打包（$H$-Subgraph-Packing）、$H$-诱导打包（$H$-Induced-Packing）、$H$-分散打包（$H$-Scattered-Packing）和 $H$-奇子式打包（$H$-Odd-Minor-Packing）问题，我们给出以下复杂度二分性：当 $H$ 为二分图时，问题在参数为二分图树宽时属于para-NP完全；而当 $H$ 为非二分图时，问题可在XP时间内求解。通过将二分图类替换为任意图类 ${\cal H}$，我们定义了1-${\cal H}$-树宽（1-${\cal H}$-treewidth）。本文开发的大多数技术适用于这一更一般的参数。