Abstract
When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.
Highlights
IntroductionAll graphs considered are assumed to be connected, unless stated otherwise
Our findings raise the question whether a k-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for k = 1 and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs
As already noticed in Chepoi and Dragan (1992); Corneil et al (2001), an algorithm breaking this quadratic barrier for general graphs is unlikely to exist since it would lead to more efficient algorithms for some disjoint set problems and, as proved in Roditty and Vassilevska Williams (2013), the latter would falsify the Strong Exponential-Time Hypothesis (SETH)
Summary
All graphs considered are assumed to be connected, unless stated otherwise. Our work completes this general result as, if we assume a total ordering over the clique to be given in the input (showing the input split graph to be k-clique-interval for some value k), it provides an almost complete characterization of the quasi linear-time solvable instances. For every nvertex k-clique-interval split graph, we can compute its diameter in quasi linear-time if k = o(log n) and a corresponding total ordering of its clique is given This result follows from an all new application of a generic framework based on k-range trees Bentley (1979), that was already used for diameter computations on some special cases Abboud et al (2016); Ducoffe (2019) but with a quite different approach than ours. Results of this paper were partially presented at the COCOA’19 conference Ducoffe et al (2019)
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