The domatic partition problem seeks to maximize the partitioning of the nodes of the network into disjoint dominating sets. These sets represent a series of virtual backbones for wireless sensor networks to be activated successively, resulting in more balanced energy consumption and increased network robustness. In this study, we address the domatic partition problem in random geometric graphs ...

We exploit the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard Optimization problems on circular arc graphs. The problems considered here are maximum domatic partition and on-line minimum vertex coloring. We present a heuristic for the domatic partition problem with a performance ratio of 4. For on-line coloring, we con...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ j for each x...

Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion–exclusion and the zeta transform. In ...

A transversal of a hypergraph is a set of vertices meeting all the hyperedges. A k-fold transversal of a hypergraph is a set of vertices such that every hyperedge has at least k elements of. In this paper, we prove that a k-fold transversal of a balanced hypergraph can be expressed as a union of k pairwise disjoint transversals and such partition can be obtained in polynomial time. We give an N...

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...

An equitable two-coloring of a hypergraph H = (V, E) is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with property B problem Radhakrishnan and Srinivasan proved that if H is a k-uniform hypergraph with maximum vertex degree ∆(H) satisfying ∆(H) c 2 k−1 √ k ln k (1) for some absolute constant c > 0, then H is 2-colorable. By using...