Digraphs+Theory,+Algorithms+and+Applications_Part3 - 1.7...

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Unformatted text preview: 1.7 Mixed Graphs and Hypergraphs 23 A mixed graph M = ( V,A,E ) contains both arcs (ordered pairs of vertices in A ) and edges (unordered pairs of vertices in E ). We do not allow loops or parallel arcs and edges, but M may have an edge and an arc with the same end-vertices. For simplicity, both edges and arcs of a mixed graph are called edges . Thus, an arc is viewed as an oriented edge (and an unoriented edge as an edge in the usual sense). A biorientation of a mixed graph M = ( V,A,E ) is obtained from M by replacing every unoriented edge xy of E by the arc xy , the arc yx or the pair xy,yx of arcs. If no unoriented edge is replaced by a pair of arcs, we speak of an orientation of a mixed graph 3 . The complete biorientation of a mixed graph M = ( V,A,E ) is a biorientation ↔ M of M such that every unoriented edge xy ∈ E is replaced in ↔ M by the pair xy,yx of arcs. Using the complete biorientation of a mixed graph M , one can easily give the definitions of a walk, trail, path, and cycle in M . The only extra condition is that every pair of arcs in ↔ M obtained in replacement of an edge in M has to be treated as two appearances of one thing. For example, if one of the arcs in such a pair appears in a trail T , then the second one cannot be in T . A mixed graph M is strong if ↔ M is strong. Similarly, one can give the definition of strong components. A mixed graph M is connected if ↔ M is connected. An edge ‘ in a connected mixed graph M is a bridge if M- ‘ is not connected. Figure 1.14 illustrates the notion of a mixed graph. The mixed graph M depicted in Figure 1.14 is strong; u, ( u,v ) ,v, { v,u } ,u is a cycle in M ; M- x has two strong components: one consists of the vertex y , the other is M = M h{ u,v,w }i ; the edge { v,w } is a bridge in M , the arc ( u,v ) and the edge { u,v } are not bridges in M ; M is bridgeless. u v w x y Figure 1.14 A mixed graph. Theorem 1.7.1 below is due to Boesch and Tindell [120]. This result is an extension of Theorem 1.6.2. We give a short proof obtained by Volkmann 3 Note that a mixed graph M = ( V,A,E ) may have a directed 2-cycle in which case no orientation of M is an oriented graph (because some 2-cycles remain). 24 1. Basic Terminology, Notation and Results [730]. (Another proof which leads to a linear time algorithm is obtained by Chung, Garey and Tarjan [157].) Theorem 1.7.1 Let e be an unoriented edge in a strong mixed graph M . The edge e can be replaced by an arc (with the same end-vertices) such that the resulting mixed graph M is strong if and only if e is not a bridge. Proof: If e is a bridge, then clearly there is no orientation of e that results in a strong mixed graph. Assume that e is not a bridge. Let M = M- e....
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Digraphs+Theory,+Algorithms+and+Applications_Part3 - 1.7...

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