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Register a new userSiblings of countable cographs
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We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomasse. The main tools are the notion of well quasi ordering and the correspondence between cographs and some labelled ordered trees.
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Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for the arboricity we want to partition the vertices into cycle-free subsets (i.e., forests). Arboricity is NP-hard in general, and our focus is on the arboricity of cographs. For arboricity two, we obtain the complete list of minimal cograph obstructions. These minimal obstructions do generalize to higher arboricities; however, we no longer have a complete list, and in fact, the number of minimal cograph obstructions grows exponentially with arboricity. We obtain bounds on their size and the height of their cotrees. More generally, we consider the following common generalization of colouring and partition into forests: given non-negative integers $p$ and $q$, we ask if a given cograph $G$ admits a vertex partition into $p$ forests and $q$ independent sets. We give a polynomial-time dynamic programming algorithm for this problem. In fact, the algorithm solves a more general problem which also includes several other problems such as finding a maximum $q$-colourable subgraph, maximum subgraph of arboricity-$p$, minimum vertex feedback set and minimum $q$ of a $q$-colourable vertex feedback set.
We characterize pairs of orthogonal countable ordinals. Two ordinals $alpha$ and $beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $alpha$ and $beta$ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if $alpha$ and $beta$ are two countable ordinals, with $alpha leq beta$, then $alpha$ and $beta$ are orthogonal if and only if either $omega + 1leq alpha$ or $alpha =omega$ and $beta < omega beta$.
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that chain complete posets whose comparability graph is a cograph are series-parallel.
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${rm gp}(G)$ of $G$. It is proved that $Ssubseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${rm diam}(G) = 2$, then ${rm gp}(G)$ is the maximum of $omega(G)$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, ${rm gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then ${rm gp}(G) leq alpha(G)$ with equality if ${rm diam}(G) in {2,3}$. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.
Let $mathrm{G}$ be a subgroup of the symmetric group $mathfrak S(U)$ of all permutations of a countable set $U$. Let $overline{mathrm{G}}$ be the topological closure of $mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $overline{mathrm{G}}[U]:={f[U]mid fin overline{mathrm{G}}}$ of images of the functions in $overline{mathrm{G}}$, being ordered under inclusion. This set $overline{mathrm{G}}[U]$ of subsets of the set $U$ will be called the emph{poset of copies for} the group $mathrm{G}$. A denomination being justified by the fact that for every subgroup $mathrm{G}$ of the symmetric group $mathfrak S(U)$ there exists a homogeneous relational structure $R$ on $U$ such that $overline G$ is the set of embeddings of the homogeneous structure $R$ into itself and $overline{mathrm{G}}[U]$ is the set of copies of $R$ in $R$ and that the set of bijections $overline Gcap mathfrak S(U)$ of $U$ to $U$ forms the group of automorphisms of $mathrm{R}$.
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