As a generalization of paths, the notion of paths of bandwidth w is introduced. We show that, for constant w ≥ 1, the corresponding search problem for such a path of length k in a given graph is NP-complete and fixed-parameter tractable (FPT) in the parameter k, like this is known for the special case w = 1, the LONGEST PATH problem. We state the FPT algorithm in terms of a guess and check prot...

The detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore de...

This is a survey of results obtained during the last 45 years regarding the intersection behaviour of all longest paths, or all longest cycles, in connected graphs. Planar graphs and graphs of higher connectivity receive special attention. Graphs embeddable in the cubic lattice of arbitrary dimension, and graphs embeddable in the triangular or hexagonal lattice of the plane are also discussed. ...

Let P be a set of n points in the plane. The geometric minimum-diameter spanning tree (MDST) of P is a tree that spans P and minimizes the Euclidian length of the longest path. It is known that there is always a monoor a dipolar MDST, i.e. a MDST with one or two nodes of degree greater 1, respectively. The more difficult dipolar case can so far only be solved in slightly subcubic time. This pap...

We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum number of edges (or vertices). These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory combined with novel dynamic programming...

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path, that is, given an undirected graph, find a maximum-length path in G. Longest Path is NP-hard in general but kno...

In this paper, we consider the minimization of the longest path length (LPL) for binary decision diagrams (BDDs) and heterogeneous multi-valued decision diagrams (MDDs). Experimental results show that: (1) For many logic functions, node minimization of BDDs also minimizes the LPLs of BDDs. (2) When we use heterogeneous MDDs for representing logic functions and minimize the memory sizes of heter...

Notation 3.1 As usual the term circuit means family of circuits, one for each n. All circuits will alternate AND and OR gates. There will be no NOT gates; however, both inputs and their negations are available. The depth of a circuit is the longest path from input to output. The fan-in of a circuit is the max number of inputs to a gate. The size of a circuits is the number of gates. A (d, f, s)...

In this paper, we propose two models of multiplication blocks by using VHDL. The algorithms that are used for writing the models are suitable for high speed multiplication and have regular cellular array structures. We have simplified some equations given in the references and then have written the VHDL model accordingly. Thus, a circuit synthesized by using the models proposed in this paper wi...

In a biconnected graph, a detour is the best alternative path from a detour-starting vertex to the destination vertex. A detour-starting vertex is the vertex from which the original shortest path is changed. The longest detour (LD) problem is to find a detour-critical edge in a shortest path such that the removal of the edge results in the maximum distance increment. In this paper, we deal with...