277 research outputs found
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 ?
A sufficient condition guaranteeing large cycles in graphs
AbstractWe generalize Bedrossian-Chen-Schelp's condition (1993) for the existence of large cycles in graphs, and give infinitely many examples of graphs which fulfill the new condition for hamiltonicity, while the related condition by Bedrossian, Chen, and Schelp is not fulfilled
On edge-sets of bicliques in graphs
A biclique is a maximal induced complete bipartite subgraph of a graph. We
investigate the intersection structure of edge-sets of bicliques in a graph.
Specifically, we study the associated edge-biclique hypergraph whose hyperedges
are precisely the edge-sets of all bicliques. We characterize graphs whose
edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its
2-section) by means of a single forbidden induced obstruction, the triangular
prism. Using this result, we characterize graphs whose edge-biclique hypergraph
is Helly and provide a polynomial time recognition algorithm. We further study
a hereditary version of this property and show that it also admits polynomial
time recognition, and, in fact, is characterized by a finite set of forbidden
induced subgraphs. We conclude by describing some interesting properties of the
2-section graph of the edge-biclique hypergraph.Comment: This version corrects an error in Theorem 11 found after the paper
went into prin
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
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