Twin-Distance-Hereditary Digraphs

TL;DR

This paper introduces directed twin-distance-hereditary graphs, a class with recursive structure enabling efficient algorithms for problems like graph coloring and width parameters, improving over previous non-recursive classes.

Contribution

It defines a new class of directed graphs with recursive construction, characterizes it via forbidden subdigraphs, and demonstrates polynomial-time algorithms for several NP-hard problems.

Findings

Directed path-width and tree-width computable in linear time.

Directed chromatic number computable in polynomial time.

Graph problems expressible in monadic second-order logic are solvable in polynomial time.

Abstract

We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. 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 the 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 which is 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.