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.

The concept of fuzzy local \(\omega\)-language and Büchi fuzzy local \(\omega\)-language are defined in \([1,2]\). In this paper, we define Landweber fuzzy local \(\omega\)-language and study their closure properties and also give an automata characterization for it. Finally, we conclude the hierarchy among the subclasses of fuzzy regular \(\omega\)-languages.

In this paper, we have calculated the combinatorial counting relations varying over the \(3\)-vertex paths of a simple graph \(G\), by restricting our attention to \(C_3\), \(C_4\)-free graphs.

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 itself is NP-complete for a general digraph. But in this paper, we solve the strong kernel problem of certain oriented networks in polynomial time.

A double shell is defined to be two edge-disjoint shells with a common apex. In this paper, we prove that double shells (where the shell orders are \(m\) and \(2m+1\)) with exactly two pendant edges at the apex are \(k\)-graceful when \(k=2\). We extend this result to double shells of any order \(m\) and \(\ell\) (where \(m \geq 3\) and \(\ell \geq 3\)) with exactly two pendant edges at the apex.

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding \((\pi, p)\) of a graph consists of a linear ordering \(\pi\) of vertices, called the spine ordering, along the spine of a book and an assignment \(p\) of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices \(u, v, x, y\) with \(\pi(u) < \pi(x) < \pi(v) 2\) and \(C_n\) are given. If \(G\) is any graph, an upper bound for the page number of the Mycielski of \(G\) is given. When \(G\) and \(H\) are any two graphs with page number \(k\) and \(l\), it is proved that the amalgamation of \(G\) and \(H\) can be embedded in a \max(k, l)\) pages. Further, we remark that the amalgamation of \(G\) with itself requires the same number of pages as \(G\), irrespective of the vertices identified in the two copies of \(G\), to form an amalgamation.

In 2004, Blinco et al [1] introduced \(\gamma\)-labeling. A function \(h\) defined on the vertex set of a graph \(G\) with \(n\) edges is called a \(\gamma\)-labeling if:

1. \(h\) is a \(p\)-labeling of \(G\),
2. \(G\) admits a tripartition \((A, B, C)\) with \(C = \{c\}\) and there exists \(\overline{b} \in B\) such that \((\overline{b}, c)\) is the unique edge with the property that \(h(c) – h(\overline{b}) = n\),
3. for every edge \(\{a, v\} \in E(G)\) with \(a \in A\), \(h(a) < h(v)\).

In [1] they have also proved a significant result on graph decomposition that if a graph \(G\) with \(n\) edges admits a \(\gamma\)-labeling then the complete graph \(K_{2cn+1}\) can be cyclically decomposed into \(2cn + 1\) copies of the graph \(G\), where \(c\) is any positive integer.

Motivated by the result of Blinco et al [1], in this paper, we prove that the well-known almost-bipartite graph, the grid with an additional edge, \((P_m \Box P_n) + \hat{e}\), admits \(\gamma\)-labeling. Further, we discuss a related open problem.

A set \( S \) of vertices of a graph \( G(V,E) \) is a \emph{dominating set} if every vertex of \( V \setminus S \) is adjacent to some vertex in \( S \). A dominating set is said to be \emph{efficient} if every vertex of \( V \setminus S \) is dominated by exactly one vertex of \( S \). A paired-dominating set is a dominating set whose induced subgraph contains at least one perfect matching. A set \( S \) of vertices in \( G \) is a total dominating set of \( G \) if every vertex of \( V \) is adjacent to some vertex in \( S \). In this paper, we construct a minimum paired dominating set and a minimum total dominating set for the infinite diamond lattice. The total domatic number of \( G \) is the size of a maximum cardinality partition of \( V \) into total dominating sets. We also demonstrate that the total domatic number of the infinite diamond lattice is 4.

In this paper we introduce right angle path and layer of an array. We construct Kolakoski array and study some combinatorial proper-ties of Kolakoski array. Also we obtain recurrence relation for layers and special elements.

An \emph{eternal 1-secure} set of a graph \(G = (V, E)\) is defined as a set \(S_0 \subseteq V\) that can defend against any sequence of single-vertex attacks by means of single guard shifts along edges of \(G\). That is, for any \(k\) and any sequence \(v_1, v_2, \ldots, v_k\) of vertices, there exists a sequence of guards \(u_1, u_2, \ldots, u_k\) with \(u_i \in S_{i-1}\) and either \(u_i = v_i\) or \(u_iv_i \in E\), such that each set \(S_i = (S_{i-1} -\{u_i\}) \cup \{v_i\}\) is dominating. It follows that each \(S_i\) can be chosen to be an eternal 1-secure set. The \emph{eternal 1-security number}, denoted by \(\sigma_1(G)\), is defined as the minimum cardinality of an eternal 1-secure set. This parameter was introduced by Burger et al. [3] using the notation \(\gamma_\infty\). The \emph{eternal \(m\)-security} number \(\sigma_m(G)\) is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an \emph{eternal \(m\)-secure} set. It was observed that \(\gamma(G) \leq \sigma_m(G) \leq \beta(G)\). In this paper, we obtain specific values of \(\sigma_m(G)\) for certain classes of graphs, namely circulant graphs, generalized Petersen graphs, binary trees, and caterpillars.