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On clique‐inverse graphs of graphs with bounded clique number
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2020-01-28 , DOI: 10.1002/jgt.22544
Liliana Alcón 1, 2 , Sylvain Gravier 3 , Claudia L. Sales 4 , Fabio Protti 5 , Gabriela Ravenna 1, 2
Affiliation  

The clique graph K (G ) of G is the intersection graph of the family of maximal cliques of G . For a family F of graphs, the family of clique‐inverse graphs of F , denoted by K 1 ( F ) , is defined as K 1 ( F ) = { H | K ( H ) F } . Let F p be the family of K p ‐free graphs, that is, graphs with clique number at most p  − 1, for an integer constant p  ≥ 2. Deciding whether a graph H is a clique‐inverse graph of F p can be done in polynomial time; in addition, for p { 2 , 3 , 4 } , K 1 ( F p ) can be characterized by a finite family of forbidden induced subgraphs. In Protti and Szwarcfiter, the authors propose to extend such characterizations to higher values of p . Then a natural question arises: Is there a characterization of K 1 ( F p ) by means of a finite family of forbidden induced subgraphs, for any p  ≥ 2? In this note we give a positive answer to this question. We present upper bounds for the order, the clique number, and the stability number of every forbidden induced subgraph for K 1 ( F p ) in terms of p .

中文翻译:

关于有界数字的图的集团逆图

所述集团图Kģ的)ģ是家庭的最大派系的交叉图形ģ。对于一个家庭 F 图,是图的集团反图 F ,表示为 ķ - 1个 F ,定义为 ķ - 1个 F = { H | ķ H F } 。让 F p 是家族ķ p -free图,即,具有至多团数曲线p  - 1,为一个整数常量p  ≥2决定的图表是否ħ是的团逆曲线图 F p 可以在多项式时间内完成;另外,对于 p { 2 3 4 } ķ - 1个 F p 可以通过有限的禁止诱导子图族来表征。在Protti和Szwarcfiter中,作者建议将这些特征扩展到p的较高值。然后一个自然的问题出现了: ķ - 1个 F p 通过的手段有限家庭禁止诱导子图,对于任何p  ≥2?在本说明中,我们对这个问题给出了肯定的答案。我们给出了每个禁止的诱导子图的阶数,集团数和稳定性数的上限 ķ - 1个 F p p表示
更新日期:2020-01-28
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