Set Representations of Linegraphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A family $\mathcal{S}$ of nonempty sets $\{S_1,\dots ,S_n\}$ is a set representation of $G$ if there exists a one-to-one correspondence between the vertices $v_1,\dots ,v_n$ in $V(G)$ and the sets in $\mathcal{S}$ such that $v_i{v}_j\in E(G)$ if and only if $S_i\cap S_j\neq \es$. A set representation $\mathcal{S}$ is a distinct (respectively, antichain, uniform and simple) set representation if any two sets $S_i$ and $S_j$ in $\mathcal{S}$ have the property $S_i\ne S_j$ (respectively, $S_i⊈S_j$, $|S_i|=|S_j|$ and $|S_i\cap S_j|⩽1$). Let $U(\mathcal{S})={\bigcup }_{i=1}^n{S}_i$. Two set representations $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ are isomorphic if ${\mathcal{S}}^{\prime }$ can be obtained from $\mathcal{S}$ by a bijection from $U(\mathcal{S})$ to $U({\mathcal{S}}^{\prime })$. Let $F$ denote a class of set representations of a graph $G$. The type of $F$ is the number of equivalence classes under the isomorphism relation. In this paper, we investigate types of set representations for linegraphs. We determine the types for the following categories of set representations: simple-distinct, simple-antichain, simple-uniform and simple-distinct-uniform.