Abstract
Let {mathcal {C}} and {mathcal {D}} be hereditary graph classes. Consider the following problem: given a graph Gin {mathcal {D}}, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to {mathcal {C}}. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:the graphs in {mathcal {C}} are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in {mathcal {D}} admit balanced separators of size governed by their density, e.g., {mathcal {O}}(varDelta ) or {mathcal {O}}(sqrt{m}), where varDelta and m denote the maximum degree and the number of edges, respectively; andthe considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes {mathcal {C}} and {mathcal {D}}:a largest induced forest in a P_t-free graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time, for every fixed t; anda largest induced planar graph in a string graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time.
Highlights
Many optimization problems in graphs can be expressed as follows: given a graph G, find a largest vertex set A such that G[A], the subgraph of G induced by A, satisfies some property
As long as the graphs from C can be recognized in polynomial time, the problem can be solved in 2n ⋅ nO(1) time by brute-force; we are interested in non-trivial improvements over this approach
Our contribution We identify three properties that together provide a way to solve the Max Induced C-Subgraph problem on graphs from D in subexponential time, where C and D are hereditary graph classes
Summary
Many optimization problems in graphs can be expressed as follows: given a graph G, find a largest vertex set A such that G[A], the subgraph of G induced by A, satisfies some property. The existence of balanced separators of size O(√m) is known for string graphs, which are intersection graphs of arc-connected subsets of the plane, and more generally for intersection graphs of connected subgraphs in any proper minor-closed class (see Lee[25]) All these observations yield a number of concrete corollaries to our main result, which are gathered in Sect. 5, we discuss some lower bounds: we show that if C is the class of forests (corresponding to the Feedback Vertex Set problem) and D is characterized by a single excluded induced subgraph, under the Exponential Time Hypothesis one cannot hope for subexponential-time algorithms in greater generality than provided by our main result
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