An activity network is an acyclic graph with non-negative weights and with a unique source and destination. A project consisting of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network provides useful information for managing the project. In a traditional activity network, it is assumed that an activity always be...

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...

Previously the authors characterized the 3-connected graphs with a Hamilton path containing only two contractible edges. In this paper we extend this result by showing that if a 3-connected graph has a diameter containing only two contractible edges, then that diameter is a Hamilton path. INTRODUCTION AND TERMINOLOGY All graphs in this paper are finite, undirected and simple. Let G be a 3-conne...

Let τ(G) denote the number of vertices in a longest path in a graph G = (V , E). A subset K of V is called a Pn-kernel of G if τ(G[K ]) ≤ n−1 and every vertex v ∈ V \K is adjacent to an end-vertex of a path of order n− 1 in G[K ]. It is known that every graph has a Pn-kernel for every positive integer n ≤ 9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possibl...

Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. In this paper, we present an approximation algorithm that generates a tree whose diameter is no more than (1+ ) times that of a GMDST, for any > 0. Our algorithm reduces the problem to several grid-aligned versions of the...

A tree is called a caterpillar if all its leaves are adjacent to the same its path, and the path is called a spine of the caterpillar. Broersma and Tuinstra proved that if a connected graph G satisfies σ2(G) ≥ |G| − k + 1 for an integer k ≥ 2, then G has a spanning tree having at most k leaves. In this paper we improve this result as follows. If a connected graph G satisfies σ2(G) ≥ |G|−k+1 and...

insert ← 1 some preprocessing to use algorithm for (i ← 1; i ≤ n; i ← i + 1) rearrange columns to put matrix in usable form for (j ← 1; j ≤ m; j ← j + 1) sorts columns by source node if (N [i, j] = 1) N ← swapcol(N, j, insert) insert ← insert + 1 top ← 1 now sorts by destination node for arcs with same source insert ← 1 while (top ≤ n) for (i ← 1; i ≤ n; i ← i + 1) for (j ← 1; j ≤ m; j ← j + 1)...

Traditional placement algorithms for FPGAs are normally carried out on a fixed clustering solution of a circuit. The impact of clustering on wirelength and delay of the placement solutions is not well quantified. In this paper, we present an algorithm named SCPlace that performs simultaneous clustering and placement to minimize both the total wirelength and longest path delay. We also incorpora...

Let P and Q be disjoint polygons in the plane. We consider problems of finding an optimal bridge (p, q), p ∈ ∂P and q ∈ ∂Q, such that the length of the longest path from a point in P , passing through the bridge (p, q), to a point Q is minimized. We propose efficient algorithms for three problems according to whether P and Q are convex or not. The algorithms developed can be easily extended to ...

TSP(1,2) is the problem of nding a tour with minimum length in a complete weighted graph where each edge has length 1 or 2. Let d o satisfy 0 < d o < 1=2. We show that TSP(1,2) has no PTAS on the set of instances where the density of the subgraph spanned by the edges with length 1 is bounded below by d o. We also show that LONGEST PATH has no PTAS on the set of instances with density bounded be...