Abstract
A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.
Highlights
The main concept studied in this paper is that of a minimal separator in a graph
Many problems that are NP-hard for general graphs become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. This is the case for Treewidth and Minimum Fill-In [12], for Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewidth at most a constant t [23], and for Distance-d Independent Set for even d [38]
We examine the consequences of these results for tame graph classes and, as our main result regarding operations preserving tame graph classes, show that the problem of determining if a hereditary graph class G is tame can be reduced to the same problem on the subclass consisting of the graphs in G that have no clique cutsets
Summary
The main concept studied in this paper is that of a minimal separator in a graph. Given a graph G, a minimal separator in G is a set of vertices that separates some non-adjacent. The result of Fomin and Villanger from [23] was further generalized in 2015 by Fomin, Todinca, and Villanger [22] to an algorithmic metatheorem concerning induced subgraphs with properties expressible in a certain logical system Their approach captures many problems including Maximum Induced Matching, Longest Induced Path, Maximum Induced Subgraph with no Cycles of Length 0 Modulo m where m is any fixed positive integer, and Maximum F-Minor-Free Induced Subgraph where F is any set of graphs containing a planar graph. All these results make it important to identify classes of graphs with a polynomially bounded number of minimal separators. The main purpose of this paper is to further the knowledge of tame graph classes, and we do this from three interrelated points of view
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