Johann Hagauer

Cartesian graph factorization at logarithmic cost per edge

LetG be a connected graph withn vertices andm edges. We develop an algorithm that finds the (unique) prime factors ofG with respect to the Cartesian product inO(m logn) time andO(m) space. This shows that factoringG is at most as costly as sorting its edges. The algorithm gains its efficiency and practicality from using only basic properties of product graphs and simple data structures.

Recognizing median graphs in subquadratic time

Abstract Motivated by a dynamic location problem for graphs, Chung, Graham and Saks introduced a graph parameter called windex. Graphs of windex 2 turned out to be, in graph-theoretic language, retracts of hypercubes. These graphs are also known as median graphs and can be characterized as partial binary Hamming graphs satisfying a convexity condition. In this paper an O(n 3 2 log n) algorithm is presented to recognize these graphs. As a by-product we are also able to isometrically embed median graphs in hypercubes in O(m log n) time.

Recognizing binary Hamming graphs inO(n2 logn) time

A graphG is called a binary Hamming graph if each vertex ofG can be assigned a binary address of fixed length such that the Hamming distance between two addresses equals the length of a shortest path between the corresponding vertices. It is shown thatO(n2 logn) time suffices for deciding whether a givenn-vertex graphG is a binary Hamming graph, and for computing a valid addressing scheme forG provided it exists. This is not far from being optimal asn addresses of lengthn — 1 have to be computed in the worst case.

Computing equivalence classes among the edges of a graph with applications

Abstract For two edges e =( x , y ) and e ′=( x ′, y ′) of a connected graph G =( V , E ) let e Θ e ′ iff d ( x , x ′)+ d ( y , y ′)≠ d ( x , y ′)+ d ( x ′, y ). Here d ( x , y ) denotes the lenght of a shortest path in G joining vertices x and y . An algorithm is presented that computes the equivalence classes induced on E by the transitive closure ΘJ of Θ in time O(| V | | E |) and space O(| V | 2 ). Finding the equivalence classes of Θ is the primary step of several graph algorithms.

Three-clustering of points in the plane

Abstract Given n points in the plane, we partition them into three classes such that the maximum distance between two points in the same class is minimized. The algorithm takes O( n 2 log 2 n ) time.

Skeletons, recognition algorithm and distance matrix of quasi-median graphs

A quasi-median graph can be characterized as a weak retract of a Cartesian product of complete graphs or equivalently as a graph of finite windex. We derive a new characterization of quasi-median graphs which allows us to recognize these graphs in O(n√n log n+m log n) time. It is shown that skeletons of quasi-median graphs are median graphs. For an arbitrary vertex s an s-skeleton of a graph is obtained from G by deleting all edges uv that satisfy d(u, s) = d(v, s). As a by-product of this approach the size of a maximum complete subgraph of a quasi-median graph can be computed within the same time bound. Furthermore, we show that the distance matrix of a quasi-median graph can be computed in O(n 2).

Three-Clustering of Points in the Plane

Given n points in the plane, we partition them into three classes such that the maximum distance between two points in the same class is minimized. The algorithm takes O(n2 log2n) time.

Recognizing binary Hamming graphs in O ( n 2 log n ) time

A graph G is called a binary Hamming graph if each vertex of G can be assigned a binary address of fixed length such that the Hamming distance between two addresses equals the length of a shortest path in G between the corresponding vertices.

Clique-gated graphs

Abstract Clique-gated graphs form an extension of quasi-median graphs. Two characterizations of these graphs are given and some other structural properties are obtained. An O( nm ) algorithm is presented which recognizes clique-gated graphs. Here n and m denote the numbers of vertices and edges of a given graph, respectively.

It is decidable whether a regular language is pure context-free

Abstract Given a regular language L we construct a pure context-free grammar G such that L is pure context-free if and only if L = L(G). Since the problem “L = L(G)?” is decidable for a regular language L and a pure context-free grammar G, it is decidable whether a regular language is pure context-free.