A self‐stabilizing algorithm for constructing a maximal (σ,τ)‐directed acyclic mixed graph

Abstract

A ( , )-directed acyclic mixed graph (DAMG) is a mixed graph, which allows both arcs (or directed edges) and (undirected) edges such that there exist exactly source nodes and 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 incident arc or both outgoing and incoming arcs. This article considers maximal ( , )-DAMG constructions: when an arbitrary undirected connected graph G = (V, E) and two distinct subsets S and T of node set V, where |S|= and |T|= , are given, construct a maximal ( , )-DAMG with source node set S and sink node set T by assigning directions to as many edges as possible (ie, by changing edges into arcs). The maximality implies that changing any more edges to arcs violates the conditions of a ( , )-DAMG (eg, a sink node has an outgoing arc or a directed cycle is created). As a previous work, a self-stabilizing algorithm for constructing a maximal (1,1)-DAMG in an arbitrary undirected connected graph is proposed for the case of = = 1. In this article, we consider construction of a maximal ( , )-DAMG for any and . First, we introduce a self-stabilizing algorithm for a maximal (1,2)-DAMG construction in any connected graph (with few constraints), which is based on the previous work. Concerning generalization of and to arbitrary values, we first clarify the necessary and sufficient condition under which a ( , )-DAMG can be constructed in which a source and a sink node sets are given. Then, we propose a generalized self-stabilizing algorithm that constructs a ( , )-DAMG when a given graph with a source and a sink node sets satisfies the above condition.

Brief Announcement: Self-stabilizing Construction of a Minimal Weakly $$\mathcal {ST}$$-Reachable Directed Acyclic Graph

“…Kim et al considered construction of another type of DAG called (σ, τ )-directed acyclic mixed graph (DAMG) [14,15,12,13] where σ and τ are the numbers of sender nodes and target nodes, respectively 1 . The reachability provided by a DAMG depends on σ and τ : strong reachability for σ = 1 and τ = 1 or 2, and weak reachability for any σ and any τ .…”

“…By weakening the reachability requirement from an ST -reachable DAG, the algorithm can construct such a DAG for any numbers of sender nodes and target nodes. The time complexity of the proposed algorithm is O(D), where D is the diameter of a given graph, which is faster than [13].…”

“…: In each cell, n, D, and ∆ are the total number of nodes, the diameter of a given graph, and the maximum degree of the graph. : The algorithm [13] has a necessary condition to construct a weakly ST -reachable DAG. * 6…”

“…* All the sender and target nodes have global, unique identifiers, while the other nodes are anonymous. * The algorithm[13] has a necessary condition to construct a weakly ST -reachable DAG. * 6…”