刻画弦 $\llcorner$-EPG 图：通过可容许团树定向
Characterizing chordal $\llcorner$-EPG graphs via admissible clique tree orientations
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中文总结 AI 辅助
本文给出连通弦 $\llcorner$-EPG 图的两种等价刻画，分别基于有序团划分树和可容许部分定向，并证明强弦性不足，提供反例。
中文摘要 AI 辅助
一个 $k$-弯路径是一条不自相交的折线，位于网格上，由至多 $k+1$ 条轴平行线段组成。一个 $\llcorner$-EPG 图是一个图，其顶点可以用网格上的 1-弯路径表示，其中每条路径要么是 $\llcorner$、要么是 $\shortmid$、要么是 $\text{-}$，使得两个顶点相邻当且仅当对应的路径共享至少一条网格边。刻画弦 $\llcorner$-EPG 图被 Cameron、Chaplick 和 Hoàng 明确地作为一个开放问题提出。我们通过给出连通弦 $\llcorner$-EPG 图的两种等价刻画来解决这个问题。第一种刻画为每个连通弦 $\llcorner$-EPG 图分配一个其顶点集的有序团划分树，其中底层树是水平和垂直边相交分量的二分关联图。第二种刻画完全用极大团表述：一个连通弦图是 $\llcorner$-EPG 图当且仅当某个团树允许一个可容许的部分定向。一个约简引理将局部团标签替换为极大团，从而将两种刻画联系起来。对于给定的团树，可容许部分定向的存在性可归约为 2-SAT，并在 $O(nM^2)$ 时间内判定，其中 $n$ 和 $M$ 分别是顶点数和极大团数。最后，我们证明强弦性并不充分：我们展示了一个强弦图，它是 $\llcorner$-EPG 图的最小禁止诱导子图，以及一个具有唯一团树的裂图族，其属于 $\llcorner$-EPG 图的条件归结为一个邻域条件。
英文摘要
A $k$-bend path is a non-self-intersecting polyline that lies on a grid and consists of at most $k+1$ axis-parallel line segments. An $\llcorner$-EPG graph is a graph whose vertices can be represented by 1-bend paths on a grid, where each path is either $\llcorner$, $\shortmid$, or $\text{-}$, such that two vertices are adjacent if and only if the corresponding paths share at least one grid edge. Characterizing chordal $\llcorner$-EPG graphs was explicitly posed as an open problem by Cameron, Chaplick, and Hoàng. We resolve this by giving two equivalent characterizations of connected chordal $\llcorner$-EPG graphs. The first assigns to every connected chordal $\llcorner$-EPG graph an ordered clique partition tree of its vertex set, where the underlying tree is the bipartite incidence graph of the horizontal and vertical edge-intersection components. The second is stated purely in terms of maximal cliques: a connected chordal graph is $\llcorner$-EPG if and only if some clique tree admits an admissible partial orientation. A reduction lemma replaces local clique labels by maximal cliques, linking the two characterizations. For a prescribed clique tree, the existence of an admissible partial orientation reduces to 2-SAT and is decided in $O(nM^2)$ time, where $n$ and $M$ are the numbers of vertices and maximal cliques. Finally, we show that strong chordality is not sufficient: we exhibit a strongly chordal graph that is a minimal forbidden induced subgraph for $\llcorner$-EPG, and a family of split graphs with a unique clique tree for which membership in $\llcorner$-EPG reduces to a neighborhood condition.
发表机构
- Lanzhou University（兰州大学）
- Huaibei Normal University（淮北师范大学）
机构由 AI 辅助整理，请以论文原文为准。
