SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 362-375, January 2021.
We unify two popular graph classes, strongly chordal graphs and chordal bigraphs, by introducing an umbrella class that contains both classes and maintains their essential properties. This is done by allowing loops at vertices. Considering loops often has little impact on a class of graphs; it however makes a big difference in this case. We call the new class \itstrongly chordal graphs with possible loops. When all vertices have loops, we recover the usual strongly chordal graphs; when all vertices are loopless, we obtain the usual chordal bigraphs. Moreover, there is a surprizing wealth of graphs in the new class that have loops at some vertices and not at others. These graphs also admit the elegant algorithms previously only applied in the extreme two cases. Formulated in the language of adjacency matrices, we study the class of symmetric 0, 1 matrices that admit a simultaneous row and column permutation avoiding the $\Gamma$ matrix $[ \begin{smallmatrix} 1 \ 1 \\ 1 \ 0 \end{smallmatrix}]$. We give ordering characterizations, matrix characterizations, and forbidden subgraph characterizations of the new class, and illustrate its usefulness by solving the minimum domination problem in this general context. This implies solutions of both the minimum dominating set in strongly chordal graphs and the minimum total dominating set in chordal bigraphs.

中文翻译：

---

具有可能循环的图的强和弦

SIAM 离散数学杂志，第 35 卷，第 1 期，第 362-375 页，2021 年 1 月。
我们通过引入一个包含两个类并保持其基本属性的伞形类来统一两个流行的图类，即强弦图和弦双图。这是通过在顶点允许循环来完成的。考虑循环通常对一类图影响不大；然而，它在这种情况下有很大的不同。我们称新类为具有可能循环的弦图。当所有顶点都有环时，我们恢复通常的强弦图；当所有顶点都是无环的，我们得到通常的和弦双图。此外，新类中有大量令人惊讶的图，它们在某些顶点具有循环，而在其他顶点则没有。这些图也承认以前仅适用于极端两种情况的优雅算法。用邻接矩阵的语言表述，我们研究对称 0 的类，1 允许同时进行行和列置换的矩阵，避免了 $\Gamma$ 矩阵 $[ \begin{smallmatrix} 1 \ 1 \\ 1 \ 0 \end{smallmatrix}]$。我们给出了新类的排序特征、矩阵特征和禁止子图特征，并通过解决这个一般上下文中的最小支配问题来说明它的用处。这意味着强和弦图中的最小支配集和和弦双图中的最小总支配集的解。并通过解决此一般上下文中的最小支配问题来说明其有用性。这意味着强和弦图中的最小支配集和和弦双图中的最小总支配集的解。并通过解决此一般上下文中的最小支配问题来说明其有用性。这意味着强和弦图中的最小支配集和和弦双图中的最小总支配集的解。