$1$ -完全可定向$K_4$ -无子图和外平面图
$1$-perfectly orientable $K_4$-minor-free and outerplanar graphs
摘要：
一个图$G$称为$1$ -完全可定向的，如果它有一个定向使得对于每个顶点$v\in V(G)$，$D$中的$v$的外邻域是$G$中的团。在$1982$中，Skrien提出了刻画$1$ -完全可定向图类的问题.这个图形类形成弦和圆弧图形类的共同概括;然而，虽然多项式可识别通过减少到$2$ -SAT，没有结构表征这类有趣的图形是已知的。在将$1$ -完全可定向图的研究归结为双连通图的基础上，利用禁止诱导子图和合成定理刻画了$1$ -完全可定向$K_4$ -无子图和$1$ -完全可定向外平面图的类.作为我们方法的一部分，我们引入了一类与$2$树类类似定义的图，并将所考虑的图类与文献中研究的在诱导子式下封闭的另外两个图类联系起来：循环可定向图和最多可分离的图~ $2$。
英文摘要：
A graph $G$ is said to be $1$-perfectly orientable if it has an orientation such that for every vertex $v\in V(G)$, the out-neighborhood of $v$ in $D$ is a clique in $G$. In $1982$, Skrien posed the problem of characterizing the class of $1$-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to $2$-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of $1$-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of $1$-perfectly orientable $K_4$-minor-free graphs and of $1$-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of $2$-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most~$2$.
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