, Compute a maximum matching for
, Compute a maximum matching for
, M , then discard M ? {v}. Otherwise, discard M
, We do so only if M satisfies some properties, SPLIT and MATCH" [101] to some module M and its neighbourhood N G (M )
, We introduce the reduction rules below and we prove their correctness. Reduction rules. The following lemma generalizes a well-known reduction rule for Maximum Matching: add a pending vertex and its unique neighbour to the matching then remove this edge, vol.78
, Lemma 19. Let M be a module in a graph G = (V, E) such that N G (M ) = {v}, F M is a maximum matching of G[M ] and F * M is obtained from F M by adding an edge between v and any unmatched vertex of M (possibly, F * M = F M if it is a perfect matching
It implies that a maximum matching F of G \ (u, v) is the union of any maximum matching of G[M \ u] with any maximum matching of G, Suppose there exists u ? M \ V (F M ). Then, u is a pending vertex of G M . There exists a maximum matching of G M ,
, v k?1 ) are vertices of G. In this situation, since N G (v 1 subgraph, with v 3 being its unique neighbour. Furthermore, by Lemma 19 we can compute a maximum matching of G[S] from F Mv x , by adding an edge between v 2 (if it is present) and an unmatched vertex in M x (if any). So, we again apply the reduction rule of Lemma 19, this time to S. Doing so, we discard S, and possibly v 3 . Then, by a symmetrical argument we can also discard M v k , M y , v k?1 and possibly v k?2 . We are left with computing a maximum matching for some subpath of
For every M we try to apply the SPLIT and MATCH technique of Definition 2. Overall, we claim that the above procedure can be implemented to run in constant time per loop. Indeed, assume that the matched vertices (resp., the unmatched vertices) are stored in a list in such a way that all the vertices in a same module M v , v ? V (G ) are consecutive. For every matched vertex u, we can access to the vertex that is matched with u in constant time. Furthermore for every v ? V (G ), we keep a pointer to the first and last vertices of M v in the list of matched vertices (resp., in the list of unmatched vertices). For any loop of the procedure, we iterate over four modules M , that is a constant. Furthermore, since |N G (M )| ? |V (G) \ M | ? 2 then we only need to check three unmatched vertices of V \ M in order to decide whether we can perform a MATCH operation. Note that we can skip scanning the unmatched vertices in M using our pointer structure, so, Case G is a spiked p-chain P k . By Theorem 24, the nontrivial modules of M(G) can only be M v 1 ,
V (F k?2 ) ? V (F max ) by construction and v k/2 (the only vertex of P k?2 possibly unmatched) is adjacent to every vertex of U . Therefore, if the subclaim were false then u 1 , u 2 should be adjacent, hence they should have been matched together with a MATCH operation, Let F max be the matching so obtained. By the above claim it takes O(|F max | ? |F 0 |)-time to compute it with the above procedure. Furthermore, we claim that F max is maximum, vol.13 ,
we should have increased the matching with a SPLIT operation ,
, By Theorem 24, v 2i ? V (G), hence M v 2i?1 is a pending module of G i . Thus, we can apply the reduction rule of Lemma 19. Doing so, we can discard the set S i , where S i = M v 2i?1 ? {v 2i } if F Mv 2i?1 is not a perfect matching of G, Case G is a spiked p-chain Q k . For every 1 ? i ? k/2 , let V i = j?i (M v 2j, with v 2i being its unique neighbour
, Then, there are two subcases. If v 2i ? S i then M z 2i?1 is disconnected in G
We apply the reduction rule of Lemma 19. Doing so, we can discard the set T i , where References, M z 2i?1 is a pending module of G[V i \ S i ] with v 2i being its unique neighbour, pp.1681-1697 ,
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, For an arbitrary graph G exactly one of the following conditions is satisfied. 1. G is disconnected, Theorem, vol.27
, 2. G is disconnected
, There is a unique proper separable p-connected component of G, with its separation being (V 1 , V 2 ) such that every vertex not in this component is adjacent to every vertex of V 1 and nonadjacent to every vertex of V 2
, G is, pp.p-connected
, If G or G is disconnected then it corresponds to a degenerate node in the modular decomposition tree. So we know how to handle with the two first cases
, A graph G is a p-tree if one of the following conditions hold: ? the quotient graph G of G is a P 4, Definition, vol.3
, ? the quotient graph G of G is a spiked p-chain P k , or its complement. Furthermore, G is obtained from G by replacing any of x, y, v 1
, ? the quotient graph G of G is a spiked p-chain Q k , or its complement. Furthermore, G is obtained from G by replacing any of v 1
, A p-connected component of a (q, q ? 3)-graph either contains less than q vertices, or is isomorphic to a prime spider, Theorem, vol.28, pp.p-tree
We use the following characterization of separable p-connected components. Theorem 29 ( [76]). A p-connected graph G is separable if and only if its quotient graph is a split graph. Furthermore, its unique separation (V 1 , V 2 ) is given by the union V 1 of the strong modules inducing the clique and the union V 2 of the strong modules inducing the stable set ,
, By Theorem 27 there are two cases. First we assume G to be p-connected. By Theorem 28, G either contains less than q vertices, or is isomorphic to a prime spider, to a disc or to a p-tree. Furthermore, if G is a p-tree then according to Definition 3, the nontrivial modules can be characterized. So, we are done in this case. Otherwise, G is not p-connected. Let V = V 1 ? V 2 ? V 3 such that: H = G[V 1 ? V 2 ] is a separable p-component with separation