上图中对$(s,1)$ -极性的最小阻碍
Minimal obstructions to $(s,1)$-polarity in cographs
摘要：
设$k,l$是非负整数。一个图$G$是$(k,l)$ -极的，如果它的顶点集允许一个划分$(A,B)$，使得$A$导出一个至多有$k$部分的完全多部图，而$B$导出一个至多有$l$团且没有其它边的不交并.如果一个图不包含$P_4$作为导出子图，则该图是上图。对于$k$和$l$的任意固定选择，$(k,l)$ -极余图可以通过有限族的禁止导出子图来刻画.确定$k = 2 = l$的确切成员的问题由Ekim，Mahadev和de Werra提出，最近由Hell，Linhares-Sales和本文第二作者解决。到目前为止，对于$0 \le k,l \le 2$，这种禁止的导出子图的完整列表是已知的;特别注意，$(1,1)$ -极图是精确分裂图。在本文中，我们主要研究$(s,1)$ -极余图的这个问题.作为我们的主要结果，我们给出了$(s,1)$ -极余图的禁用导出子图的递归完全刻画，对于每个非负整数$s$。此外，我们证明了具有$(s,1)$ -分区的某些整数$s$（这里$s$是不固定的）的上图可以通过禁止一个家庭的四个图。
英文摘要：
Let $k,l$ be nonnegative integers. A graph $G$ is $(k,l)$-polar if its vertex set admits a partition $(A,B)$ such that $A$ induces a complete multipartite graph with at most $k$ parts, and $B$ induces a disjoint union of at most $l$ cliques with no other edges. A graph is a cograph if it does not contain $P_4$ as an induced subgraph. It is known that $(k,l)$-polar cographs can be characterized through a finite family of forbidden induced subgraphs, for any fixed choice of $k$ and $l$. The problem of determining the exact members of such family for $k = 2 = l$ was posted by Ekim, Mahadev and de Werra, and recently solved by Hell, Linhares-Sales and the second author of this paper. So far, complete lists of such forbidden induced subgraphs are known for $0 \le k,l \le 2$; notice that, in particular, $(1,1)$-polar graphs are precisely split graphs. In this paper, we focus on this problem for $(s,1)$-polar cographs. As our main result, we provide a recursive complete characterization of the forbidden induced subgraphs for $(s,1)$-polar cographs, for every non negative integer $s$. Additionally, we show that cographs having an $(s,1)$-partition for some integer $s$ (here $s$ is not fixed) can be characterized by forbidding a family of four graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
