The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where $$\pi(G)=\lceil q-\frac{g}{g-2}(p-2)\rceil$$ , and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.

中文翻译：

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π-skew 图的性质与应用

用 sk(G) 表示的图 G 的偏度是 G 中去除其导致平面图的边的最小数量。它是衡量图与平面性的接近程度的一个重要参数，它是最大平面子图问题的补充，并且在计算上等效。对于 p 顶点和 q 边上周长为 g 的任何连通图 G，可以很容易地验证 sk(G) ≥ π(G)，其中 $$\pi(G)=\lceil q-\frac{g}{g -2}(p-2)\rceil$$ ，如果等式成立，则图 G 被称为 π 偏斜。π-skew 的概念首先由 GL Chia 和 CL Lee 提出。周长为 3 的 π 偏斜图正是包含三角剖分作为生成子图的图。本文的目的是探索 π-skew 图的性质。通过应用这些属性，可以获得一些 π 偏斜图族，包括两个图的连接、完全多部图和两个图的笛卡尔积。我们还讨论了存在跨越三角剖分的阈值。在其他结果中，给出了关于图的规则性和大小的一些充分条件，这些条件确保了跨越三角剖分。