Consider the following problem, that we call “Chordless Path through Three Vertices” or CP3V, for short: Given a simple undirected graph G = (V,E), a positive integer k, and three distinct vertices s, t, and v ∈ V , is there a chordless path from s via v to t in G that consists of at most k vertices? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternat...

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P . The kPC problem is ...

The longest (s,t)-path problem on supergrid graphs is known to be NP-complete. However, the complexity of this with or without holes still unknown.In past, we presented linear-time algorithms for solving L-shaped and C-shaped graphs, which form subclasses holes. In paper, will determine O-shaped a subclass These are rectangular It worth noting that contain as subgraphs, but there no inclusion r...

With the help of recursion relations derived from the self-similar structure, we obtain the solution of average path length, d[over ]_(t) , for Apollonian networks. In contrast to the well-known numerical result d[over ]_{t} proportional, variant(ln N_(t));(3/4) [J. S. Andrade, Jr., Phys. Rev. Lett. 94, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmical...

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by triangular tessellation. In order to evaluate how well tessellation approximates space, we three types paths: path~$ \mathit{SP_w}(s,t) $, which path from $ s t in space; vertex \mathit{SVP_w}(s,t) where vertices are tessellation; and grid \mathit{SGP_w}(s,t) whose edges The ratios \frac{\lVe...

Here C is the power of the continuum and a path z(t ; (o), 0 < t < o; ;, is said to have a point of multiplicity C if there exist a point ~ and a set T of positive numbers having the power of the continuum such that z(t; (o) = S for all t e T. This result, combined with those of the previous papers [1] and [3], completely settles the question of points of highest multiplicity of Brownian paths ...