Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs

A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms

Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes \oplus\L (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in \FL^{\SPL}. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.Comment: 23 pages, 13 figure

Algoritma matching bobot maskimum dalam graph bipartit komplit berboto

ABSTRAK Suatu matching dalam graph G adalah subgraph 1-regular pada G yang disebabkan oleh kumpulan dart pasangan garis yang tidak adjacent. Suatu matching merupakan matching maksimum bila matching tersebut mempunyai harga pokok maksimum. Matching dalam graph bipartit merupakan matching maksimum apabila tidak adanya path perluasart yang berkenaan dengan matching tersebut. Matching yang mempunyai bobot maksimum disebut matching bobot maksimum. Matching bobot maksimum dalam graph bipartit komplit berbobot diperoleh dengan mencari matching maksimum dalam subgraph pada graph bipartit komplit berbobot, kemudian dibangun sampai didapatkan matching perfek atau setiap titik dalam V merupakan titik matched. A matching in a graph G is a 1-regular subgraph of G, that is, a subgraph induced _by a collection of pairwise nonadjacent edges. A matching is called maximum matching if the matching have maximum cardinality. A matching in a bipartite graphs is a maximum matching if there exists no augmenting path. A matching in which the sum of the weights of maximum its edges is called maximum weight matching. A maximum weight matching in weighted complete bipartite graphs is got to find maximum matching in subgraph to weighted complete bipartite graphs, further its construct to arrived is got perfec matching or each vertex in V is matched vertex