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 \( (\overrightarrow{u, v}) \) 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 for an undirected graph \( G \) and solve it in polynomial time for certain cycle-related 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 is NP-complete for a general digraph. In this paper, we solve the strong kernel problem of an oriented biregular graph in polynomial time.

String-token Petri net, which is a variation of coloured Petri net, has been introduced in [1] by requiring the tokens to be labeled by strings. Languages in regular and linear families, which are two basic classes in the Chomsky hierarchy, are generated by these Petri nets [2]. An extension called array-token Petri net, introduced in [5] by labeling tokens by arrays, generates picture languages. Properties related to generative power of array-token Petri net are considered in [3]. In this paper, application of array-token Petri net to generate English alphabetic letters treated as rectangular arrays is examined.

Given a graph \( G = (V, E) \), a set \( W \subseteq V \) is said to be a resolving set if for each pair of distinct vertices \( u, v \in V \), there is a vertex \( x \) in \( W \) such that \( d(u, x) \neq d(v, x) \). The resolving number of \( G \) is the minimum cardinality of all resolving sets. In this paper, a condition is imposed on resolving sets and a conditional resolving parameter is studied for grid-based networks.

Let \( G = (V, E) \) be a graph. A vertex labeling \( f: V \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E \). For each \( i \in \mathbb{Z}_2 \), define \( v_f(i) = |f^{-1}(i)| \) and \( e_f(i) = |{f^*}^{-1}(i)| \). We call \( f \) friendly if \( |v_f(1) – v_f(0)| \leq 1 \). The full friendly index set of \( G \) is the set of all possible values of \( e_f(1) – e_f(0) \), where \( f \) is a friendly labeling. In this paper, we study the full friendly index set of the wheel \( W_n \), the tensor product of paths \( P_2 \) and \( P_n \), i.e., \( P_2 \otimes P_n \), and the double star \( D(m, n) \).

The detour order of a graph \( G \), denoted \( \tau(G) \), is the order of a longest path in \( G \). A partition \( (A, B) \) of \( V(G) \) such that \( \tau(\langle A \rangle) \leq a \) and \( \tau(\langle B \rangle) \leq b \) is called an \( (a, b) \)-partition of \( G \). A graph \( G \) is called \( \tau \)-\textit{partitionable} if \( G \) has an \( (a, b) \)-partition for every pair \( (a, b) \) of positive integers such that \( a + b = \tau(G) \).

The well-known Path Partition Conjecture states that every graph is \( \tau \)-partitionable. Motivated by the recent result of Dunbar and Frick [6] that if every \( 2 \)-connected graph is \( \tau \)-partitionable, then every graph is \( \tau \)-partitionable, we show that the Path Partition Conjecture is true for a large family of \( 2 \)-connected graphs with certain ear-decompositions. Also, we show that a family of \( 2 \)-edge-connected graphs with certain ear-decompositions is \( \tau \)-partitionable.

he problem of determining the collaboration graph of co-authors of Paul Erdos is a challenging task. Here we take up this problem for the case of Rolf Nevanlinna Prize Winners. Even though the number of prize winners as of date is 7, the collaboration graph has 20 vertices and 41 edges and possesses several interesting properties. In this paper, we have obtained this graph and determined standard graph parameters for the graph as well as its complement besides probing its structural properties. Several new results were obtained.

The topological descriptor Wiener index, named after the chemist Harold Wiener, is defined as half the sum of the distances between every pair of vertices of a graph. A lot of research has been devoted to finding the Wiener index by brute force method. In this paper, we compute the Wiener index of chemical structures such as sodium chloride and benzenoid without using a distance matrix.

A graph \( G(p, q) \) is said to be total edge bimagic with two common edge counts \( k_1 \) and \( k_2 \) if there exists a bijection \( f: V(G) \cup E(G) \to \{1, 2, \ldots, p + q\} \) such that for each edge \( uv \in E(G) \), \( f(u) + f(v) + f(uv) = k_1 \) or \( k_2 \).
A total edge-bimagic graph is called super edge-bimagic if \( f(V(G)) = \{1, 2, \ldots, p\} \). In this paper, we define new types of super edge-bimagic labeling and prove some interesting results related to super edge-bimagic labeling. Also, its relationship with cordial labeling is studied.

Trust is one of the most important means to improve the reliability of computing resources provided in a cloud environment and it plays an important role in commercial cloud environments. Trust is the estimation of the capability of a cloud resource in completing a task based on reputation, identity, behavior, and availability in the context of a distributed environment. It helps customers in the selection of appropriate resources in heterogeneous cloud infrastructure. The cloud computing depends on the following QoS parameters such as reliability, availability, scalability, security, and past behavior of the cloud resources.
This paper introduces a novel trust model to evaluate cloud resources of IaaS (Infrastructure as a Service) providers by means of Trust Resource Broker. The Trust Resource Broker selects trustworthy cloud resources based on the requirements of customers. The proposed trust model evaluates the trust value of the resources based on the identity as well as behavioral trust. The proposed model applies the QoS metrics suitable for cloud resources. The results of the experiments show that the proposed trust model selects the most reliable resources in a cloud environment.