Polar graphs generalise bipartite, cobipartite, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we sho...

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for 2P3-free chordal graphs while it is solvable in polynomial time for P...

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance of at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-colorings with optimal or near-optimal λ arises in the context of radio frequency assignment. We show that the problem of ...

The notion of a split coloring of a complete graph was introduced by Erdős and Gyárfás [7] as a generalization of split graphs. In this paper, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two-round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)-split coloring problem...

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-coloring with small or optimal λ arises in the context of radio frequency assignment. We show that the problem of finding the...

In this note by considering a complete lattice L, we define thenotion of an L-Fuzzy hyperrelation on a given non-empty set X. Then wedefine the concepts of (POM)L-Fuzzy graph, hypergraph and subhypergroupand obtain some related results. In particular we construct the categories ofthe above mentioned notions, and give a (full and faithful) functor form thecategory of (POM)L-Fuzzy subhypergroups ...

We consider the following question: can split graphs with odd maximum degree be edge-coloured with maximum degree colours? We show that any odd maximum degree split graph can be transformed into a special split graph. For this special split graph, we were able to solve the question, in case the graph has a quasi-universal vertex.

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 chord...

We present a fully dynamic algorithm for split graphs that supports the following types of operations: (1) query whether deleting or inserting an edge preserves the split property, (2) query whether inserting a new vertex with a given neighborhood in the current graph preserves the split property, (3) insert or delete an edge or a vertex when the split property is preserved, (4) insert an edge ...