$2$-极性及cograph超类上极性变体的算法方面

Neighborhood covering and independence on two superclasses of cographs

Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v \in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex $v$. Two elements of $E(G) \cup V(G)$ are neighborhood-independent if there is no vertex $v\in V(G)$ such that both elements are in $G[v]$. A set $S\subseteq V(G)\cup E(G)$ is neighborhood-independent if every pair of elements of $S$ is neighborhood-independent. Let $ρ_{\mathrm n}(G)$ be the size of a minimum neighborhood cover set and $α_{\mathrm n}(G)$ of …

Minimal obstructions for polarity, monopolarity, unipolarity and $(s,1)$-polarity in generalizations of cographs

It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of $P_4$-reducible graphs. In this work, we prove that also when restricted to the classes of $P_4$-sparse graphs and $P_4$-extendible graphs (both of which extend $P_4$-reducible graphs) every hereditary property can be characterized by finitely many minimal obstructions. We present complete lists of $P_4$-sparse and $P_4$-extendible minimal obstructions for polarity, monopolarity, unipolarity, and $(s,1)$-polarity, where $s$ is a p…

Minimal obstructions to $(\infty, k)$-polarity in cographs

A graph is a cograph if it does not contain a 4-vertex path as an induced subgraph. An $(s, k)$-polar partition of a graph $G$ is a partition $(A, B)$ of its vertex set such that $A$ induces a complete multipartite graph with at most $s$ parts, and $B$ induces the disjoint union of at most $k$ cliques with no other edges. A graph $G$ is said to be $(s, k)$-polar if it admits an $(s, k)$-polar partition. The concepts of $(s, \infty)$-, $(\infty, k)$-, and $(\infty, \infty)$-polar graphs can be analogously defined. Ekim, Mahadev and de Werra pioneered in the research on polar cographs, obtaining…

Minimal obstructions to $(s,1)$-polarity in cographs

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…

$\mathcal F$-cographs 的量子性质

Quantum properties of $\mathcal F$-cographs

本文系统研究有限图的量子不对称、量子对称与量子同构，提出 Schmidt 备选，解决 cographs 与 forests 的相关问题并计算量子自同构群，还推广至其严格超类并证明平面图量子同构即同构。

Minimal obstructions to $2$-polar cographs

A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ into parts $A$ and $B$ such that the subgraph induced by $A$ is a complete multipartite graph with at most $k$ parts, and the subgraph induced by $B$ is a disjoint union of at most $k$ cliques with no other edges. It is known that $k$-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed $k$. A concrete family of such forbidden induced subgraphs is known for $k=1$, since $1$-polar graphs are precisely split graphs. For larger…