Journal of Algorithms and Computation
Volume:48 Issue: 1, Dec 2016

We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the n! Hamilton cycles in a complete directed graph on n + 1 vertices corresponds with each of the n! n-permutation matrices P, such that pu</sub>,i</sub> = 1 if and only if the ith</sup> arc in a cycle enters vertex u, starting and ending at vertex n + 1. A graph instance (G) is initially coded as exclusion set E, whose members are pairs of components of P, {pu,i</sub>, pv</sub>,i+1</sub>}, i = 1, n - 1, for each arc (u, v) not in.the formula is not displayed correctly!

A special class of cubic graphs are the cycle permutation graphs. A cycle permutation graph Pn</sub></em>(α</em>) is defined by taking two vertex-disjoint cycles on n</em> vertices and adding a matching between the vertices of the two cycles.In this paper we determine a good upper bound for tenacity of cycle permutation graphs.the formula is not displayed correctly!

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f (u), f(v)). f is called k-prime cordial labeling of G if |vf</sub> (i) − vf</sub> (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef</sub> (0) − ef</sub> (1)| ≤ 1 where vf</sub> (x) denotes the number of vertices labeled with x, ef</sub> (1) and ef</sub> (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 3- prime cordial labeling behavior of union of a 3-prime cordial graph and a path Pn</sub>.the formula is not displayed correctly!

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*</sup>: V (G) → Z - {0} defined by f*</sup>(v) = ΣP∈</span>Ev</sub> f (e) is one-one where Ev</sub> denotes the set of edges in G that are incident with a vertex v and f*</sup>(V (G)) is either of the form {±k1</sub>,±k2</sub>,...,±kp</sub>/2</sub>} or {±k1</sub>,±k2</sub>,...,±k(p-1)</sub>/2</sub>} U {±k(p+1)</sub>/2</sub>} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the graphs GL(n), double triangular snake D(Tn</sub>), Wn</sub>, Fln</sub>, <Cm</sub>,K1,n and <Cm</sub> * K1,n admit edge pair sum labeling.the formula is not displayed correctly!

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf</sub> (i) − vf</sub> (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef</sub> (0) − ef</sub> (1)| ≤ 1 where vf</sub> (x) denotes the number of vertices labeled with x, ef</sub> (1) and ef</sub> (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 4- prime cordial labeling behavior of complete graph, book, flower, mCn</sub> and some more graphs.the formula is not displayed correctly!

Let G (V, E) be a graph with p vertices and q edges. A graph G is said to have an odd mean labeling if there exists a function f : V (G) → {0, 1, 2, . . . , 2q − 1} satisfying f is 1 − 1 and the induced map f ∗ : E(G) → {1, 3, 5, . . . , 2q − 1} defined by f ∗ (uv) = ( f(u)+f(v) 2 if f(u) + f(v) is even f(u)+f(v)+1 2 if f(u) + f(v) is odd is a bijection. A graph that admits an odd mean labeling is called an odd mean graph. In this paper, we have studied an odd meanness property of the subdivision of the slanting ladder SLn for all n ≥ 2, CnK1 for n ≥ 3, the grid Pm ×Pn for m, n ≥ 2, Cm @Cn for m, n ≥ 3 and P2m  nK1 for all m, n ≥ 1..the formula is not displayed correctly!

In this paper, we present a binary-linear optimization model to prevent the spread of an infectious disease in a community. The model is based on the remotion of some connections in a contact network in order to separate infected nodes from the others. By using this model we nd an exact optimal solution and determine not only the minimum number of deleted links but also their exact positions. The formulation of the model is insensitive to the number of edges in a graph and can be used (with complete or local information) to measure the resistance of a network before and after an infectious spreads. Also, we propose some related models as generalizations: quarantining problem including resource constraints (time, budget, etc.), maximum rescued nodes-minimum deleted links problem and minimum removed links problem nding a prespecied number of nodes with weakest connections.

A graceful labeling of a graph G</em> of size n</em> is an injective assignment of integers from {0, 1,..., n</em>} to the vertices of G</em>, such that when each edge of G</em> has assigned a weight, given by the absolute dierence of the labels of its end vertices, the set of weights is {1, 2,..., n</em>}. If a graceful labeling f</em> of a bipartite graph G</em> assigns the smaller labels to one of the two stable sets of G</em>, then f</em> is called an -labeling and G</em> is said to be an α</em>-graph. A tree is a caterpillar if the deletion of all its leaves results in a path. In this work we study graceful labelings of the disjoint union of a cycle and a caterpillar. We present necessary conditions for this union to be graceful and, in the case where the cycle has even size, to be an α</em>-graph. In addition, we present a new family of graceful trees constructed using α</em>-labeled caterpillars.the formula is not displayed coorectly!

A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted by tvs(G)is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the total vertex irregularity strength for n ≥ 3, m ≥ 2, Pn ⊙ K1, Pn ⊙ K2, Cn ⊙ K2, Ln ⊙ K1, CLn ⊙ K1, P2 ⊙ Cn, Pn ⊙ Km, Cn ⊙ Km.the formula is not displayed correctly!

In this paper we discuss about tenacity and its properties in stability calculation. We indicate relationships between tenacity and connectivity, tenacity and binding number, tenacity and toughness. We also give good lower and upper bounds for tenacity.

If we think of the graph as modeling a network, the vulnerability measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. Many graph theoretical parameters have been used to describe the vulnerability of communication networks, including connectivity, integrity, toughness, binding number and tenacity.
In this paper we discuss tenacity and its properties in vulnerability calculation.

The analysis of vulnerability in networks generally involves some questions about how the underlying graph is connected. One is naturally interested in studying the types of disruption in the network that maybe caused by failures of certain links or nodes. In terms of a graph, the concept of connectedness is used in dierent forms to study many of the measures of vulnerability. When certain vertices or edges of a connected graph are deleted, one wants to know whether the remaining graph is still connected, and if so, what its vertex - or edge - connectivity is. If on the other hand, the graph is disconnected, the determination of the number of its components or their orders is useful. Our purpose here is to describe and analyze the current status of the vulnerability measures, identify its more interesting variants, and suggest a most suitable measure of vulnerability.