an equitable domination has interesting application in the contextof social networks. in a network, nodes with nearly equal capacitymay interact with each other in a better way. in the societypersons with nearly equal status, tend to be friendly. in thispaper, we introduce new variant of equitable domination of agraph. basic properties and some interesting results have beenobtained.

An equitable k-partition of a graph G is collection induced subgraphs (G[V1],G[V2],…,G[Vk]) such that (V1,V2,…,Vk) partition V(G) and −1≤|Vi|−|Vj|≤1 for all 1≤i<j≤k. We prove every planar admits an 2-partition into 3-degenerate graphs, 3-partition 2-degenerate two forests one graph.

We consider two kinds of partition of a graph, namely orbit partitions and equitable partitions. Although an orbit partition is always an equitable partition, the converse is not true in general. We look at some classes of graphs for which the converse is true.

Positional analysis is considered an important tool in the analysis of social networks. It involves partitioning of the set of actors into subsets such that actors in a subset are similar in their structural relationships with other actors. Traditional methods of positional analysis such as structural equivalence, regular equivalence, and equitable partitions are either too strict or too generi...

We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of partitions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minim...

The Kuramoto model describes the collective dynamics of a system of coupled oscillators. An α-Kuramoto partition is a graph partition induced by the Kuramoto model, when the oscillators include a phase frustration parameter. We prove that every equitable partition is an α-Kuramoto partition, but that the converse does necessarily not hold. We give an exact characterisation of α-Kuramoto biparti...

We study how the number c(X) of components of a graph X can be expressed through the number and properties of the components of a quotient graph X/∼ . We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of co...

There is a vast literature already in existence on colourings in graphs and designs. We refer the interested reader to [2, 7, 8, 11, 12]. A major application of such colourings is to sampling and scheduling problems. For an excellent consideration of designs for statistical purposes, see [13]; in [8], examples of graph colouring applications in scheduling are described. Let P be a point set and...