8 research outputs found
Parameterized Leaf Power Recognition via Embedding into Graph Products
The k-leaf power graph G of a tree T is a graph whose vertices are the leaves of T and whose edges connect pairs of leaves at unweighted distance at most k in T. Recognition of the k-leaf power graphs for k >= 6 is still an open problem. In this paper, we provide an algorithm for this problem for sparse leaf power graphs. Our result shows that the problem of recognizing these graphs is fixed-parameter tractable when parameterized both by k and by the degeneracy of the given graph. To prove this, we describe how to embed the leaf root of a leaf power graph into a product of the graph with a cycle graph. We bound the treewidth of the resulting product in terms of k and the degeneracy of G. As a result, we can use methods based on monadic second-order logic (MSO_2) to recognize the existence of a leaf power as a subgraph of the product graph
Pairwise Compatibility Graphs: A Survey
International audienceA graph is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and two nonnegative real numbers and such that each leaf of is a node of and there is an edge if and only if , where is the sum of weights of the edges on the unique path from to in . In this article, we survey the state of the art concerning this class of graphs and some of its subclasses
A forbidden induced subgraph characterization of distance-hereditary . . .
AbstractA graph G is a k-leaf power if there is a tree T such that the vertices of G are the leaves of T and two vertices are adjacent in G if and only if their distance in T is at most k. In this situation T is called a k-leaf root of G. Motivated by the search for underlying phylogenetic trees, the notion of a k-leaf power was introduced and studied by Nishimura, Ragde and Thilikos and subsequently in various other papers. While the structure of 3- and 4-leaf powers is well understood, for k≥5 the characterization of k-leaf powers remains a challenging open problem.In the present paper, we give a forbidden induced subgraph characterization of distance-hereditary 5-leaf powers. Our result generalizes known characterization results on 3-leaf powers since these are distance-hereditary 5-leaf powers