Abstract: A chord of a path P is an edge joining two non-adjacent vertices of P . A path P is called a monophonic path if it is a chordless path. A monophonic graphoidal cover of a graph G is a collection ψm of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The minimum card...

We present a new technique called Multiple Vertex Next Event Estimation, which outperforms current direct lighting techniques in forward scattering, optically dense media with the Henyey-Greenstein phase function. Instead of a one-segment connection from a vertex within the medium to the light source, an entire sub path of arbitrary length can be created and we show experimentally that 4-10 seg...

A path from a source vertex v to a target vertex u is said to be a shortest path if its total cost is minimum among all v-to-u paths. Dijkstra’s algorithm is a classic shortest path algorithm, which is described in many textbooks. To justify its correctness (whose rigorous proof will be given in the next article), it is necessary to clarify its underlying principle. For this purpose, the articl...

1. INTRODUCTION. A Hamiltonian cycle in a graph is a path in the graph which visits each vertex exactly once and returns to the starting vertex. Let í µí°¾ í µí±› be a weighted complete graph with í µí±› vertices. We define the weight of an edge as the square of the distance between two end points of the edge. The weight of a path í µí±ƒ is the sum of the weights of all edges in the path and de...

A path in an edge colored graph is said to be a rainbow path if every edge in this colored with the same color. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertexconnection number of G denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainb...

In this paper we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. This is the second paper in a series of two. The first one, [3] is also submitted to this journal. In [2, 3], a polynomial time algorithm is given for three-coloring triangle-free graphs with no i...

Agraph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a ...

A tournament is a digraph, where there is precisely one arc between every pair of distinct vertices. An arc is pancyclic in a digraph D, if it belongs to a cycle of length l, for all 3 <= l <= |V (D)|. Let p(D) denote the number of pancyclic arcs in a digraph D and let h(D) denote the maximum number of pancyclic arcs belonging to the same Hamilton cycle of D. Note that p(D) >= h(D). Moon showed...

We introduce graphs called hypochordal : for any path of length 2, there exists a chord or another path of length 2 between its two endpoints. We show that such graphs are 2-vertex-connected and moreover in the case of an edge or a vertex deletion, the distance between any pair of nonadjacent vertices remains unchanged. We give properties of hypochordal graphs, then we study the class of minimu...

In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent ...