The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ∪ V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ∪ V (C2...

A graph G is called k-ordered if for every sequence of k distinct vertices there is a cycle traversing these vertices in the given order. In the present paper we consider two novel generalizations of this concept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered . We prove the following results for a chordal graph G: (a) G is (2k − 3)-connected if and only if it is k-vertex-edge-ordered...

Fifty-two patients (35 female, 17 male) completed questionnaires related to their experiences during the first 7 days of appliance wear (31 fixed, 21 removable) and then after 14 and 90 days. Most problems relating to discomfort and pain resolved within 4-7 days. The problems encountered with fixed appliances were generally more severe than with removable appliances. Swallowing and speech were ...

Given a graph G = (V, E), V(G) = V and E(G) = E denote the vertex set and the edge set of G, respectively. All graphs considered in this paper are undirected graphs. A simple path (or path for short) is a sequence of adjacent edges (v1, v2), (v2, v3), ..., (vm-2, vm-1), (vm-1, vm), written as 〈v1, v2, v3, ..., vm〉, in which all of the vertices v1, v2, v3, ..., vm are distinct except possibly v1...

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow...

In this paper some new type of odd graceful graphs are investigated. We prove that the graph obtained by joining a cycle C8 with some star graphs S1,r keeping one vertex and three vertices gap between pair of vertices of the cycle admits odd graceful labelling. We also prove that the graph obtained by joining a cycle C12 with some star graphs S1,r keeping two, three and five vertices gap betwee...

Given a digraph D = (V, A), a quasi-kernel of D is an independent set Q C_ V such that for every vertex v not contained in Q, either there exists a vertex u E Q such that v dominates u, or there exists a vertex w such that v dominates w and w dominates some vertex u E Q. A sink in a digraph D = (V, A) is a vertex v E V that dominates no vertex of D. In this thesis we prove that if D is a semico...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. A strong stable set of a graph G is a stable set that intersects all maximal cliques of G. We present a linear-time algorithm that, for any vertex of a Meyniel graph, find a strong stable set containing this vertex.

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. We propose two algorithms to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithms have the advantage of finding each chordless cycle only once. To ensure this, we introduced the concepts of vertex labeling and initial valid vertex triplet. To g...