Digraphs+Theory,+Algorithms+and+Applications_Part32

Digraphs+Theory,+Algorithms+and+Applications_Part32 - 11.1...

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Unformatted text preview: 11.1 Properly Coloured Trails in Edge-Coloured Multigraphs 603 alternating cycles P = { C 1 ,...,C p } such that x and y belong to some cycles in P and Ω ( P ) is a connected graph. We formulate the following trivial but useful observation as a proposition. Proposition 11.1.12 Cyclic connectivity is an equivalence relation on the vertices of a 2-edge-coloured multigraph. ut This proposition allows us to consider cyclic connectivity components similar to strong connectivity components of digraphs. The following theorem due to A. Yeo (private communication, 1998) shows that cyclic connectivity between a pair of vertices can be checked in polyno- mial time. Theorem 11.1.13 For a pair x,y of vertices in a 2-edge-coloured multigraph H = ( V,E ) , one can check whether x and y are cyclic connected in time O ( | E | ( | V | + | E | )) . Proof: By Proposition 11.1.11, in time O ( | E | ), one can check whether H has an alternating cycle through a fixed edge e ∈ E . This implies that, in time O ( | V || E | ), one can verify whether H has an alternating cycle through a fixed vertex v ∈ V . We now describe a polynomial algorithm to check whether x and y are cyclic connected. Our algorithm starts by initiating X := { x } . Then, we find an alternating cycle through x ; let X be the vertices except for x of such a cycle. If y ∈ X , then we are done. Otherwise, delete the vertices of X from H , set X := X and X := ∅ . Then, for each edge e with one end-vertex in X and the other not in X find an alternating cycle through the edge (if one exists). Now append all the vertices, except for those in X , in the cycles we have found to X and check whether y ∈ X . If y / ∈ X , then we continue as above. We proceed until either y ∈ X or there is no alternating cycle through any edge with one end-vertex in X and the other not in X . Clearly, if y ∈ X at some stage, then x and y are cyclic connected, otherwise they are not. The total time required for the operation of deletion is O ( | V || E | ). By the complexity bounds above and the fact that we may want to find an alternating cycle through an edge at most once, the complexity of the described algorithm is O ( | E | ( | V | + | E | )). ut The following theorem by Bang-Jensen and Gutin shows that cyclic con- nectivity implies colour-connectivity. Theorem 11.1.14 [64] If a pair, x,y , of vertices in a 2-edge-coloured multi- graph G is cyclic connected, then x and y are colour-connected. Proof: If x and y belong to a common alternating cycle, then they are colour-connected. So, suppose that this is not the case. 604 11. Generalizations of Digraphs Since x and y are cyclic connected, there is a collection P = { C 1 ,...,C p } of alternating cycles in G so that x ∈ V ( C 1 ), y ∈ V ( C p ), and, for every i = 1 , 2 ,...,p- 1 and every j = 1 , 2 ,...,p , | i- j | > 1, V ( C i ) ∩ V ( C i +1 ) 6 = ∅ , V ( C i ) ∩ V ( C j ) = ∅ . ( P corresponds to a ( C 1 ,C p )-path in Ω ( R ), where R is the set of all alternating cycles in...
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Digraphs+Theory,+Algorithms+and+Applications_Part32 - 11.1...

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