Twin-distance-hereditary digraphs

Abstract

Distance-hereditary graphs are in important graph class in graph theory, as they are well-placed in the graph class hierarchy and permit many algorithmic results. We investigate structural and algorithmic advantages of a directed version of this well-researched graph class. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several twin and pendant vertex operations analogously to undirected distance-hereditary graphs and which still preserves the distance hereditary property. We give a characterization by forbidden induced subdigraphs and place the class in a hierarchy comparing it to related classes. We further show algorithmic advantages concerning directed width parameters, directed graph coloring and some other well-known digraph problems which are NP-hard in general, but computable in polynomial or even linear time on twin-distance-hereditary digraphs. This includes computability of directed path-width and tree-width in linear time and the directed chromatic number in polynomial time. From our result that directed twin-distance-hereditary graphs have directed clique-width at most 3 it follows by Courcelle's theorem on directed clique-width that we can compute every graph problem describable in monadic second-order logic on quantification over vertices and vertex sets as well as some further problems like Hamiltonian Path/Cycle in polynomial time.