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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 36 (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 15 (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