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 \( v_1, v_2, \ldots, v_p \in V \) such that \( x \to v_i, y \to v_i \) are arcs of the digraph \( D \) for each \( i = 1, 2, \ldots, p \). In this paper, we prove that double stars \( DS_m \) (\( m \geq 2 \)) are \( p \)-competition graphs. We also show that full regular \( m \)-ary trees \( T_{m,n} \) with height \( n \) are \( p \)-competition graphs, where \( p \leq \frac{m – 1}{2} \).