We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) Dominating ($f$-BDS). Let $G=(V,E)$ be a graph with color assignment function $\gamma \colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ from each ...

Given a graph G, a dominating set D is a set of vertices such that any vertex not in D has at least one neighbor in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) to prove a ‘‘Vizing-like’’ inequ...

Dominating sets are fundamental graph structures. However, enumeration of dominating sets has not received much attention. This study aims to propose two efficient enumeration algorithms for sparse graphs. The first algorithm enumerates all the dominating sets for k-degenerate graphs in O (k) time per solution using O (n + m) space. Since planar graphs have a constant degeneracy, this algorithm...

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...

Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the nec...

Fluxes of ions and water across the short-circuited, excised bullfrog lung were determined by radioisotope techniques. The unidirectional flows of Na+, K+, Ca++, TcO4 minus, HCO3 minus, gluconate, rho-aminohippurate, dinitrophenolate, SO4 equal to, and water were symmetrical. Both HCO3 minus fluxes were reduced by acetazolamide. In contrast, Cl minus, Br minus I minus, and SCN minus movement fr...

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γm, and of an m-tuple dominating set is γ×m. For a property π of subsets of V (G), wit...

Uncertainty is inherent in many important applications, such as data integration, environmental surveillance, location-based services (LBS), sensor monitoring and radio-frequency identification (RFID). In recent years, we have witnessed significant research efforts devoted to producing probabilistic database management systems, and many important queries are re-investigated in the context of un...