Characterising $(k,\ell )$-leaf powers

Abstract

We say that, for $k\ge 2$ and $\ell >k$, a tree $T$ with distance function $d_T(x,y)$ is a $(k,\ell )$-leaf root of a finite simple graph $G=(V,E)$ if $V$ is the set of leaves of $T$, for all edges $xy\in E$, $d_T(x,y)\le k$, and for all non-edges $xy\notin E$, $d_T(x,y)\ge \ell$. A graph is a $(k,\ell )$-leaf power if it has a $(k,\ell )$-leaf root. This new notion modifies the concept of $k$-leaf powers (which are, in our terminology, the $(k,k+1)$-leaf powers) introduced and studied by Nishimura, Ragde and Thilikos; $k$-leaf powers are motivated by the search for underlying phylogenetic trees. Recently, a lot of work has been done on $k$-leaf powers and roots as well as on their variants phylogenetic roots and Steiner roots. Many problems, however, remain open.

We give the structural characterisations of $(k,\ell )$-leaf powers, for some $k$ and $\ell$, which also imply an efficient recognition of these classes, and in this way we improve and extend a recent paper by Kennedy, Lin and Yan on strictly chordal graphs; one of our motivations for studying $(k,\ell )$-leaf powers is the fact that strictly chordal graphs are precisely the $(4,6)$-leaf powers.