Many standard graph classes are known to be characterized by means of layouts (a permutation of its vertices) excluding some patterns. Important such graph classes are among others: proper interval graphs, interval graphs, chordal graphs, permutation graphs, (co-)comparability graphs. For example, a graph $G=(V,E)$ is a proper interval graph if and only if $G$ has a layout $L$ such that for every triple of vertices such that $x\prec_L y\prec_L z$, if $xz\in E$, then $xy\in E$ and $yz\in E$. Such a triple $x$, $y$, $z$ is called an indifference triple and layouts excluding indifference triples are known as indifference layouts. In this paper, we investigate the concept of tree-layouts. A tree-layout $T_G=(T,r,\rho_G)$ of a graph $G=(V,E)$ is a tree $T$ rooted at some node $r$ and equipped with a one-to-one mapping $\rho_G$ between $V$ and the nodes of $T$ such that for every edge $xy\in E$, either $x$ is an ancestor of $y$ or $y$ is an ancestor of $x$. Clearly, layouts are tree-layouts. Excluding a pattern in a tree-layout is defined similarly as excluding a pattern in a layout, but now using the ancestor relation. Unexplored graph classes can be defined by means of tree-layouts excluding some patterns. As a proof of concept, we show that excluding non-indifference triples in tree-layouts yields a natural notion of proper chordal graphs. We characterize proper chordal graphs and position them in the hierarchy of known subclasses of chordal graphs. We also provide a canonical representation of proper chordal graphs that encodes all the indifference tree-layouts rooted at some vertex. Based on this result, we first design a polynomial time recognition algorithm for proper chordal graphs. We then show that the problem of testing isomorphism between two proper chordal graphs is in P, whereas this problem is known to be GI-complete on chordal graphs.

翻译：许多标准图类已知可通过排除某些模式的布局（顶点排列）来刻画。重要的此类图类包括：真区间图、区间图、弦图、置换图、（余）可比图。例如，图 $G=(V,E)$ 是真区间图当且仅当存在布局 $L$，使得对于任意满足 $x\prec_L y\prec_L z$ 的三元组，若 $xz\in E$，则 $xy\in E$ 且 $yz\in E$。这样的三元组 $x$、$y$、$z$ 称为无差异三元组，而排除无差异三元组的布局称为无差异布局。本文研究树布局的概念。图 $G=(V,E)$ 的树布局 $T_G=(T,r,\rho_G)$ 是一棵以节点 $r$ 为根的树 $T$，并配备 $V$ 与 $T$ 节点间的一一映射 $\rho_G$，使得对于每条边 $xy\in E$，要么 $x$ 是 $y$ 的祖先，要么 $y$ 是 $x$ 的祖先。显然，布局是树布局的特例。在树布局中排除模式的定义与在布局中排除模式类似，但现使用祖先关系。通过树布局排除某些模式可以定义尚未探索的图类。作为概念验证，我们证明在树布局中排除非无差异三元组给出了真弦图的自然定义。我们刻画了真弦图并将其置于已知弦图子类的层级结构中。我们还提供了真弦图的规范表示，该表示编码了所有以某个顶点为根的无差异树布局。基于此结果，我们首先设计了真弦图的多项式时间识别算法。然后证明了两个真弦图之间的同构测试问题属于P类，而该问题在弦图上已知为GI完全问题。