Unique perfect phylogeny is NP-hard

We answer, in the affirmative, the following question proposed by Mike Steel as a $100 challenge: "Is the following problem NP-hard? Given a ternary phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the only tree that displays Q ?

Reduced clique graphs of chordal graphs

AbstractWe investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. We propose a novel decomposition of reduced clique graphs based on two operations: edge contraction and removal of the edges of a split. Based on this decomposition, we characterize asteroidal sets in chordal graphs, and discuss chordal graphs that admit a tree representation with a small number of leaves

Dichotomy for tree-structured trigraph list homomorphism problems

Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP-complete. Thus while digraph list homomorphism problems enjoy dichotomy (each problem is NP-complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems. However, in this paper, we identify a large class of trigraphs for which list homomorphism problems do exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and, in particular, include all trigraphs whose underlying graphs are trees. In fact, we show that for these tree-like trigraphs, the trigraph list homomorphism problem is polynomially equivalent to a related digraph list homomorphism problem. We also describe a few examples illustrating that our conditions defining tree-like trigraphs are not unnatural, as relaxing them may lead to harder problems

On P 4 -transversals of Chordal Graphs

Abstract A P 4 -transversal of a graph G is a set of vertices T which meets every P 4 of G. A P 4 -transversal T is called stable if there are no edges in the subgraph of G induced by T . It has been previously shown by Hoàng and Le that it is N P -complete to decide whether a comparability (and hence perfect) graph G has a stable P 4 -transversal. In the following we show that the problem is N P -complete for chordal graphs. We apply this result to show that two related problems of deciding whether a chordal graph has a P 3 -free P 4 -transversal, and deciding whether a chordal graph has a P 4 -free P 4 -transversal (also known as a two-sided P 4 -transversal) are both N Pcomplete. Additionally, we strengthen the main results to strongly chordal graphs

Unique perfect phylogeny is N P -hard

Abstract. We answer, in the affirmative, the following question proposed by Mike Steel as a $100 challenge: "Is the following problem N Phard? Given a ternary † phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the only tree that displays Q?" [28, 29] As a particular consequence of this, we show that the unique chordal sandwich problem is also N P -hard

The vertex leafage of chordal graphs

Every chordal graph $G$ can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called {\em tree model} of $G$. The leafage $\ell(G)$ of a connected chordal graph $G$ is the minimum number of leaves of the host tree of a tree model of $G$. The vertex leafage \vl(G) is the smallest number $k$ such that there exists a tree model of $G$ in which every subtree has at most $k$ leaves. The leafage is a polynomially computable parameter by the result of \cite{esa}. In this contribution, we study the vertex leafage. We prove for every fixed $k\geq 3$ that deciding whether the vertex leafage of a given chordal graph is at most $k$ is NP-complete by proving a stronger result, namely that the problem is NP-complete on split graphs with vertex leafage of at most $k+1$. On the other hand, for chordal graphs of leafage at most $\ell$, we show that the vertex leafage can be calculated in time $n^{O(\ell)}$. Finally, we prove that there exists a tree model that realizes both the leafage and the vertex leafage of $G$. Notably, for every path graph $G$, there exists a path model with $\ell(G)$ leaves in the host tree and it can be computed in $O(n^3)$ time