Note on $p$-Competition Graphs of Double Stars

Abstract

The $p$-competition graph $C_p(D)$ of a digraph $D=(V,A)$ is a graph with $V(C_p(D))=V(D)$, where an edge between distinct vertices $x$ and $y$ if and only if there exist $p$ distinct vertices $v_1,v_2,\dots ,v_p\in V$ such that $x\to v_i,y\to v_i$ are arcs of the digraph $D$ for each $i=1,2,\dots ,p$. In this paper, we prove that double stars $D{S}_m$ ($m\ge 2$) are $p$-competition graphs. We also show that full regular $m$-ary trees $T_{m,n}$ with height $n$ are $p$-competition graphs, where $p\le \frac{m–1}{2}$.