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 exist 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 incident arc or both outgoing and incoming arcs. This article considers maximal (σ,τ)-DAMG constructions: when an 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 (ie, by changing edges into arcs). The maximality implies that changing any more edges to arcs violates the conditions of a (σ,τ)-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 σ=τ=1. In this article, we consider construction of a maximal (σ,τ)-DAMG for any σ and τ. 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 σ and τ to arbitrary values, we first clarify the necessary and sufficient condition under which a (σ,τ)-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 (σ,τ)-DAMG when a given graph with a source and a sink node sets satisfies the above condition.

中文翻译：

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一种构造最大（σ，τ）向非循环混合图的自稳定算法

有（σ，τ）向的非循环混合图（DAMG）是混合图，它允许圆弧（或有向边）和（无向）边都存在，从而恰好存在σ源节点和τ沉节点，但不存在定向循环（仅由弧组成）。每个源（重新接收）弧至少有一个向外（传入）弧，但没有传入（向外）弧。而且，任何其他节点既不是源节点也不是宿节点。它没有入射弧，也没有流出弧和流入弧。本文考虑了最大（σ，τ）-DAMG构造：当任意无向连通图G =（V，E）和节点集V的两个不同子集S和T，其中| S | = σ和| 给出T | = τ，通过将方向分配给尽可能多的边（即，通过将边变成弧形）来构造具有源节点集S和宿节点集T的最大（σ，τ）-DAMG 。最大值意味着将更多的边更改为弧会违反a（σ，τ-DAMG（例如，汇点节点具有向外的弧或创建了有向循环）。作为先前的工作，提出了一种用于在σ = τ = 1的情况下构造任意无向连接图中的最大（1,1）-DAMG的自稳定算法。在本文中，我们考虑针对任何σ和τ构造一个最大（σ，τ）-DAMG 。首先，我们基于以前的工作，针对任何连接图中的最大（1,2）-DAMG构造引入了一种自稳定算法（约束很少）。关于将σ和τ推广为任意值，我们首先阐明a（σ，τ）-DAMG可以被构造，其中给出了源节点和宿节点集。然后，我们提出了一种广义的自稳定算法，当给定具有源节点和宿节点的图满足上述条件时，该算法将构造（σ，τ）-DAMG。