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From arXiv

Fixed-Parameter Complexity of Minimum Profile Problems

Let G=(V,E)G=(V,E) be a graph. An ordering of GG is a bijection α:V\dom{1,2,...,V}.\alpha: V\dom \{1,2,..., |V|\}. For a vertex vv in GG, its closed neighborhood is N[v]={uV:uvE}{v}.N[v]=\{u\in V: uv\in E\}\cup \{v\}. The profile of an ordering α\alpha of GG is \prfα(G)=vV(α(v)min{α(u):uN[v]}).\prf_{\alpha}(G)=\sum_{v\in V}(\alpha(v)-\min\{\alpha(u): u\in N[v]\}). The profile \prf(G)\prf(G) of GG is the minimum of \prfα(G)\prf_{\alpha}(G) over all orderings α\alpha of GG. It is well-known that \prf(G)\prf(G) is the minimum number of edges in an interval graph HH that contains GG is a subgraph. Since V1|V|-1 is a tight lower bound for the profile of connected graphs G=(V,E)G=(V,E), the parametrization above the guaranteed value V1|V|-1 is of particular interest. We show that deciding whether the profile of a connected graph G=(V,E)G=(V,E) is at most V1+k|V|-1+k is fixed-parameter tractable with respect to the parameter kk. We achieve this result by reduction to a problem kernel of linear size.
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Published on April 24, 2006
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