Minimal Split Completions of Graphs

Abstract

We study the problem of adding edges to a given arbitrary graph so that the resulting graph is a split graph, called a split completion of the input graph. Our purpose is to add an inclusion minimal set of edges to obtain a minimal split completion, which means that no proper subset of the added edges is sufficient to create a split completion. Minimal completions of arbitrary graphs into chordal graphs have been studied previously, and new results have been added continuously. There is an increasing interest in minimal completion problems, and minimal completions of arbitrary graphs into interval graphs have been studied very recently. We extend these previous results to split graphs, and we give a characterization of minimal split completions, along with a linear time algorithm for computing a minimal split completion of an arbitrary input graph. Among our results is a new way of partitioning the vertices of a split graph uniquely into three subsets.

Characterizing and Computing Minimal Cograph Completions

“…For the second one, however, a clever implementation does not help as long as we output an explicit representation of the completed graph. All the other known algorithms that compute minimal completions in linear time [19,22,38], in fact, use some implicit representation. For cographs we can always use cotrees, but there are also other interesting representations that might be even more suitable for our problem, like the vertex ordering suggested in [9].…”

“…This fact encouraged researchers to focus on various alternatives that are computationally more efficient, at the cost of optimality or generality. Examples of the approaches that have been attempted include approximation [36], restricted input [7,6,34,29,10,28], parameterization [12,26,23,15,33] and minimal completions [19,21,22,25,38,41]. Here we consider the last alternative.…”

“…Hence, for polynomial time recognizable classes with this property, it becomes trivial to solve the characterization problem, and very easy to solve both the extraction and the computation problems as well. Examples of algorithms that exploit sandwich monotonicity for efficiently extracting and computing a minimal completion, are those for chordal [4], split [19], threshold and chain graph [22] completions. In contrast, among the classes that do not have the sandwich monotone property, the only one for which a solution to the characterization and extraction problems is known, is the class of interval graphs [24].…”

Minimal Proper Interval Completions

“…Heggernes and Mancini [7] gave a linear time algorithm for computing a minimal embedding into split graphs.…”

Single-Edge Monotonic Sequences of Graphs and Linear-Time Algorithms for Minimal Completions and Deletions

“…Here we define a threshold partition (S, K) of a threshold graph in a unique way. Note first that all vertices of degree more than ω(G) − 1 must belong to K, and all vertices of degree less than ω(G) − 1 must belong to S. The set of vertices of degree exactly ω(G) − 1 is either an independent set or a clique [18]. If this set is a clique, we place all of these vertices in K, and if it is an independent set we place them in S. We refine the sets S and K further as follows: (S 0 , S 1 , S 2 , ..., S ) is a partition of S such that S 0 is the set of isolated vertices, and…”