A Self-Stabilizing Algorithm for Constructing a Maximal (2,2)-Directed Acyclic Mixed Graph

A (σ, τ)-directed acyclic mixed graph (DAMG) is a mixed graph which allows both arcs (or directed edges) and (undirected) edges such that there exists exactly σ source nodes and τ sink nodes but there exists no directed cycle (consisting of only arcs). Each source (resp. sink) node has at least one outgoing (resp. incoming) arc but no incoming (resp. outgoing) arc. Moreover any other node is neither a source nor a sink node; it has no arc or both outgoing and incoming arcs. This paper considers maximal (σ, τ)-DAMG construction: when any arbitrary undirected connected graph G = (V, E) and two distinct subsets S and T of node set V, where |S|=σ and |T|=τ, are given, construct a maximal (σ, τ)-DAMG with source node set S and sink node set T by assigning directions to as many edges as possible (i.e., by changing edges into arcs). The maximality implies that changing any more edges to arcs violates the conditions of a (σ, τ)-DAMG (e.g., a sink node has an outgoing arc or a directed cycle is created). In this paper, at first, we propose a self-stabilizing algorithm for the maximal (1,2) -DAMG construction in any connected graph (with few constraints). Finally, we propose a self-stabilizing algorithm for the maximal (2,2) -DAMG construction problem, which is an extension of the algorithm for constructing a maximal (1,2) -DAMG.