Forbidden subgraphs in enhanced power graphs of finite groups
Xuanlong Ma, Samir Zahirović, Yubo Lv, Yanhong She
备注：
This paper has been accepted for publication in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. Note that this is not a finally version. Its finally version will soon appear in the journal's homepage
本文研究有限群的增强幂图的禁用子图相关分类问题，完成了增强幂图为split、threshold图的有限群分类，以及对应弦图、cograph的幂零群分类并给出非幂零群实例，部分解答了Peter J. Cameron的问题。
摘要：
群的增强幂图是一种简单图，其顶点集由群的所有元素组成，且任意一对顶点若生成一个循环子群则相邻。本文对所有增强幂图为split图和threshold图的有限群进行了分类；还对所有增强幂图为弦图和cograph的有限幂零群进行了分类；最后给出了若干类增强幂图为弦图和cograph的非幂零群。这些结果部分回答了Peter J. Cameron提出的一个问题。
英文摘要：
The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split and threshold. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs. These results partly answer a question posed by Peter J. Cameron.
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Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on set…
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A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to rev…
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Zero forcing is a process that models the spread of information throughout a graph as white vertices are forced to turn blue using a color change rule. The idea of throttling, introduced in 2013 by Butler and Young, is to optimize the trade-off between the number of initial blue vertices and the time taken to force all vertices to become blue. The original throttling number of a graph minimizes the sum of these two quantities and the product throttling number minimizes their product. In addition, weighted throttling changes the weights given to these two quantities when minimizing their sum. S…
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The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…
A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.
Considering connected $K_{1,3}$-free graphs with independence number at least $3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is $2ω$-colourable where $ω$ denotes the clique number of $G$. We study $(K_{1,3}, Y)$-free graphs, and show that the following three statements are equivalent. (1) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is perfect. (2) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is $ω$-colourable…
It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with some $k\in \mathbb{N}$ bounding the diameter of members of $\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs.…
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We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
