Abstract
The problem Max W-Light (Max W-Heavy) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-Light is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, Max W-Heavy can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for Max W-Light, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin et al. (SIAM J Comput 44:54---87, 2015). The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.
Highlights
IntroductionWe show that if a hereditary graph class has a polynomial upper bound on the number of potential maximal cliques and has a function depending only on degeneracy as an upper bound of treewidth, the metatheorem of Fomin et al can be applied to Max W -Light
Let G = (V, E) be an undirected graph
We show that if a hereditary graph class has a polynomial upper bound on the number of potential maximal cliques and has a function depending only on degeneracy as an upper bound of treewidth, the metatheorem of Fomin et al can be applied to Max W -Light
Summary
We show that if a hereditary graph class has a polynomial upper bound on the number of potential maximal cliques and has a function depending only on degeneracy as an upper bound of treewidth, the metatheorem of Fomin et al can be applied to Max W -Light. It is known that chordal graphs, circular-arc graphs, d-trapezoid graphs, and chordal bipartite graphs have polynomial upper bounds on the number of potential maximal cliques (see Section 4) We show that these hereditary graph classes have functions of degeneracy as upper bounds on treewidth, and our algorithms can be applied. We observe that graphs of bounded clique-width admit a function of degeneracy as an upper bounded of treewidth, and Max W -Heavy can be solved in linear time. We show that for any fixed W , Max W -Light is W[1]-complete, while Max W -Heavy admits a kernel of O(W k) vertices, where the parameter k is the solution size
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