ll graphs considered in our study are simple, finite and undirected. A graph is equitable total domination edge addition critical (stable) if the addition of any arbitrary edge changes (does not change) the equitable total domination number. In this paper, we introduce the following new parameters: equitable independent dom- ination number, equitable total domination number and equitable connected domination number and study their stability upon edge addition, on special families of graphs namely cycles, paths and com- plete bipartite graphs. Also the relation among the above parameters is established.

The \(k\)-rainbow domination is a variant of the classical domination problem in graphs and is defined as follows: Given an undirected graph \(G = (V, E)\) and a set of \(k\) colors numbered \(1, 2, \dots, k\), we assign an arbitrary subset of these colors to each vertex of \(G\). If a vertex is assigned the empty set, then the union of color sets of its neighbors must be \(k\) colors. This assignment is called the \(k\)-rainbow dominating function of \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we present some bounds on the \(3\)-rainbow domination number of circulant networks and grid networks.

A linear layout, or simply a layout, of an undirected graph \( G = (V, E) \) with \( n = |V| \) vertices is a bijective function \( \phi: V \to \{1, 2, \dots, n\} \). A \( k \)-coloring of a graph \( G = (V, E) \) is a mapping \( \kappa: V \to \{c_1, c_2, \dots, c_k\} \) such that no two adjacent vertices have the same color. A graph with a \( k \)-coloring is called a \( k \)-colored graph.

A colored layout of a \( k \)-colored graph \( (G, \kappa) \) is a layout \( \phi \) of \( G \) such that for any \( u, x, v \in V \), if \( (u, v) \in E \) and \( \phi(u) < \phi(x) < \phi(v) \), then \( \kappa(u) \neq \kappa(x) \). Given a \( k \)-colored graph \( (G, \kappa) \), the problem of deciding whether there is a colored layout \( \phi \) of \( (G, \kappa) \) is NP-complete. In this paper, we introduce the concept of chromatic layout of \( G \) and determine the chromatic layout number for paths and cycles.

Let \( G(V, E) \) be a simple graph. For a labeling \( \partial: V \cup E \to \{1, 2, 3, \dots, k\} \), the weight of a vertex \( x \) is defined as

\[
wt(x) = \partial(x) + \sum_{xy \in E} \partial(xy).
\]

The labeling \( \partial \) is called a vertex irregular total \( k \)-labeling if for every pair of distinct vertices \( x \) and \( y \), \( wt(x) \neq wt(y) \). The minimum \( k \) for which the graph \( G \) has a vertex irregular total \( k \)-labeling is called the total vertex irregularity strength of \( G \) and is denoted by \( tvs(G) \). In this paper, we obtain a bound for the total vertex irregularity strength of honeycomb and honeycomb derived networks.

Graph embedding is an important technique used in the study of computational capabilities of processor interconnection networks and task distribution. In this paper, we present an algorithm for embedding the Hypercubes into Banana Trees and Extended Banana Trees and prove its correctness using the Congestion lemma and Partition lemma.

A kernel in a directed graph \(D(V, E)\) is a set \(S\) of vertices of \(D\) such that no two vertices in \(S\) are adjacent and for every vertex \(u\) in \(V \setminus S\), there is a vertex \(v\) in \(S\) such that \((u, v)\) is an arc of \(D\). The problem of existence of a kernel is NP-complete for a general digraph. In this paper, we introduce the acyclic kernel problem of an undirected graph \(G\) and solve it in polynomial time for uniform theta graphs and even quasi-uniform theta graphs.

Given a graph \( G = (V, E) \), a labeling \( \partial: V \cup E \to \{1, 2, \dots, k\} \) is called an edge irregular total \( k \)-labeling if for every pair of distinct edges \( uv \) and \( xy \), \( \partial(u) + \partial(uv) + \partial(v) \neq \partial(x) + \partial(xy) + \partial(y) \). The minimum \( k \) for which \( G \) has an edge irregular total \( k \)-labeling is called the total edge irregularity strength of \( G \). In this paper, we examine the hexagonal network, which is a well-known interconnection network, and obtain its total edge irregularity strength.

Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. In this paper, we prove that grid and cylinder are the subgraphs of certain circulant networks. Further, we present an algorithm to embed tori into certain circulant networks with dilation\(2\) and vice-versa.

Broadcasting is a fundamental information dissemination problem in a connected graph, in which one vertex called the originator disseminates one or more messages to all other vertices in the graph. \(A\)-broadcasting is a variant of broadcasting in which an informed vertex can disseminate a message to at most \(k\) uninformed vertices in one unit of time. In general, solving the broadcast problem in an arbitrary graph is NP-complete. In this paper, we obtain the \(k\)-broadcast time of the Sierpiński gasket graphs for all \(k \geq 1\).

Let \( G = (V, E) \) be a graph with vertex set \( V \) and edge set \( E \). Let \( \text{diam}(G) \) denote the diameter of \( G \) and \( d(u, v) \) denote the distance between the vertices \( u \) and \( v \) in \( G \). An antipodal labeling of \( G \) with diameter \( d \) is a function \( f \) that assigns to each vertex \( u \), a positive integer \( f(u) \), such that \( d(u, v) + |f(u) – f(v)| \geq d \), for all \( u, v \in V \). The span of an antipodal labeling \( f \) is \( \max \{|f(u) – f(v)| : u, v \in V(G)\} \). The antipodal number for \( G \), denoted by \( \text{an}(G) \), is the minimum span of all antipodal labelings of \( G \). Determining the antipodal number of a graph \( G \) is an NP-complete problem. In this paper, we determine the antipodal number of certain graphs.