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.
翻译:(k,l)$-polar cograph 可以用一个固定的美元和美元来表示一个完整的多部分图,最多部分为美元部分,最多部分为美元部分,而美元B$则引起一个最多为美元 cliques 的脱节,而没有其他边緣。一个图表是一个暗淡的拼图,如果它不包含 P_4 美元作为导引的子谱。已知美元(k,l) 的polar cograph 可以通过一个被禁止的子集(A,B) 美元来表示(k,l) $-polar) 。对于任何固定的美元和美元选择来说,A$将引出一个完整的多部分的多部分图。对于确定这种家庭的确切成员的问题,Ekim, Mahadev和de Werra, 最近由Hell, Linhares- Sales 和本文的第二作者来解决。因此, 这样的禁止子图的完整清单可以用来表示$0, le,l\ $ lix $ 美元 。