A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...

The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3n, up to polynomial factors. In this paper, we desi...

We prove a new structural property regarding the “skyline” of uniform radius disks and use this to derive a number of new sequential and distributed approximation algorithms for well-known optimization problems on unit disk graphs (UDGs). Specifically, the paper presents new approximation algorithms for two problems: domatic partition and weighted minimum dominating set (WMDS) on UDGs, both of ...

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...

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 ∑ x∈N[v] f (x) ≥ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f1, f2, . . . , fd} of signed dominating functions on Gwith the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (G), is called a signed dominating fa...

Instead of studying evolutions governed by an evolutionary system starting at a given initial state on a prescribed future time interval, finite or infinite, we tackle the problem of looking both for a past interval [T − D, T ] of aperture (or length, duration) D and for the viable evolutions arriving at a prescribed terminal state at the end of the temporal window (and thus telescoping if more...

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.