The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application o...

Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that • lpt(G) 6 ⌈

We prove a lower bound of 32D for the longest path in arbitrary graphs that use two-label interval routing (2-IRS) as the routing method, where D is the diameter of the graph.

A new method for recognizing and locating objects by searchiiig longest paths. Abstract This paper present,s a new 111ethod for recognizing ant1 locating an ol~ject in an i~uage. T l ~ e contours of the sl~apes are tlcscril)ttl by line seg~nents. The basic idea is t o gener-at? tliago~lal paths in a t,al)le where tlre colunlns represent t.he. ~llotltl s e g i ~ ~ e l ~ t s a.11tl the rows the s...

Let T be a tree in which every edge is associated with a real number. The sum of a path in T is the sum of the numbers associated with the edges of the path and its length is the number of the edges in it. For two positive integers L1 ≤ L2 and two real numbers S1 ≤ S2, a path is feasible if its length is between L1 and L2 and its sum is between S1 and S2. We address the problem: Given a tree T ...

The following problem is considered: Given a predicate p on strings. Determine the longest segment of a given string that satisfies p. This paper is an investigation of algorithms solving this problem for various predicates. The predicates considered are expressed in simple functions like the size, the minimum, the maximum, the leftmost and the rightmost element of the segment. The algorithms a...

SUMMARY Longest path problem is a problem for finding a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, there are few known graph classes such that the longest path problem can be solved efficiently. Polynomial time algorithms for finding a longest cycle and a longest path in a Ptolemaic graph are pr...