Given a weighted directed graph G = (V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A ⊆ A such that the directed graph (V,A \ A) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple co...

In 1991, Gnanajothi [4] proved that the path graph n P with n vertex and 1 n − edge is odd graceful, and the cycle graph m C with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph m n C P ∪ when 4, 6,8,10 m = and then we proved that the graph m n C P ∪ is odd graceful if m is even. Finall...

This paper addresses the problem of locating two facilities 11 in vertex-weighted tree/cycle/unicycle networks, where each facility can 12 fail with a given probability [16]. It is assumed that the two facilities 13 do not fail simultaneously, and when a facility fails, the other facility is 14 required to service all the vertices. We show that the back-up 2-center on 15 a weighted path (resp. ...

The pebbling number f(G) of a graphG is the least p such that, no matter how p pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H , we have f(G × H) ≤ f(G)f(H). If the graph G satisfies the odd 2-pebbling property, we w...

It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords...

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We show that such graphs are precisely those in which the induced subgraph, namely, a cycle with exactly one chord is forbidden. We also show that deciding maxim...

an impacted tooth can be performed with either fixed or removable appliances. Fixed appliances do not require special patient cooperation and can produce precise coronal and root movements.1,2 On the other hand, the wire attached to the impacted tooth may be more easily deflected than with removable appliances, causing undesirable movements of adjacent teeth.3 Such movements are minimized with ...

UNLABELLED While utilizing removable dentures, regardless of whether the dentures are acrylic or fabricated of eclipse resin, the need for their filling-relining or their repair frequently arises. The purpose of this study is to display the technicalities of the procedure for rebasing and relining of removable light curing eclipse denture resin. MATERIAL AND METHODS This study presents cases ...

This paper addresses the existence of Hamiltonian paths and cycles in two-dimensional grids consisting of triangles or quadrilaterals, and three-dimensional grids consisting of tetrahedra or hexahedra. The paths and cycles may be constrained to pass from one element to the next through an edge, through a vertex, or be unconstrained and pass through either. It was previously known that an uncons...

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. A special case of L-cycle covers are k-cycle covers for k ∈ N, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We com...