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Parameterized graph separation problems
 In Proc. 1st IWPEC, volume 3162 of LNCS
, 2004
"... We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminal ..."
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Cited by 54 (3 self)
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We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are
Parameterized coloring problems on chordal graphs
 Theor. Comput. Sci
, 2006
"... In the precoloring extension problem (PrExt) a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper coloring of the graph with the given number of colors. Two parameterized versions of the problem are studied in ..."
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Cited by 17 (4 self)
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In the precoloring extension problem (PrExt) a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper coloring of the graph with the given number of colors. Two parameterized versions of the problem are studied in the paper: either the number of precolored vertices or the number of colors used in the precoloring is restricted to be at most k. We show that for chordal graphs these problems are polynomialtime solvable for every fixed k, but W[1]hard if k is the parameter. For a graph class F, let F + ke (resp., F +kv) denote those graphs that can be made to be a member of F by deleting at most k edges (resp., vertices). We investigate the connection between PrExt in F (with the two parameters defined above) and the coloring of F + ke, F + kv graphs (with k being the parameter). Answering an open question of Leizhen Cai [5], we show that coloring chordal+ke graphs is fixedparameter tractable. 1
Fast Hamiltonicity checking via bases of perfect matchings
, 2012
"... For an even integer t ≥ 2, the Matching Connectivity matrix Ht is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph Kt on t vertices; an entry Ht[M1,M2] is 1 if M1 ∪M2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Ham ..."
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Cited by 12 (1 self)
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For an even integer t ≥ 2, the Matching Connectivity matrix Ht is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph Kt on t vertices; an entry Ht[M1,M2] is 1 if M1 ∪M2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Hamiltonicity problem, we present three results on the structure ofHt: We first show thatHt has rank at most 2t/2−1 over GF(2) via an appropriate factorization that explicitly provides families of matchings Xt forming bases for Ht. Second, we show how to quickly change representation between such bases. Third, we notice that the sets of matchings Xt induce permutation matrices within Ht. Subsequently, we use the factorization to obtain an 1.888nnO(1) time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Our algorithm as well counts the number of Hamiltonian cycles modulo two in directed bipartite or undirected graphs in the same time bound. Moreover, we use the fast basis change algorithm from the second result to present a Monte Carlo algorithm that given an undirected graph on n vertices along with a path decomposition of width at most pw decides Hamiltonicity in (2 + 2)pwnO(1) time. Finally, we use the third result to show that for every > 0 this cannot be improved to (2 + 2 − )pwnO(1) time unless the Strong Exponential Time Hypothesis fails, i.e., a faster algorithm for this problem would imply the breakthrough result of a (2 − )n time algorithm for CNFSat. 1