Janez Zerovnik

Estimating the traffic on weighted cactus networks in linear time

A communication network can be modeled by a graph with weighted vertices and edges corresponding to the amount of traffic from sources and expected delays at links. We give a linear algorithm for computing the sum of all delays on a weighted cactus graphs. Cactus is a graph in which every edge lies on at most one cycle. The sum of delays is equivalent to the weighted Wiener number, a well known graph invariant in mathematical chemistry. Complexity of computing Wiener polynomial on cacti is discussed.

Finding a five bicolouring of a triangle-free subgraph of the triangular lattice

A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. Often the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of size p(υ) to each vertex of the triangular lattice so that the sets of colours assigned to adjacent vertices are disjoint. A n-[p]colouring of a graph G is a mapping c from V(G) into the set of the subsets of {1,2,...,n} such that |c(υ)|=p(υ) and for any adjacent vertices u and υ, c(u)∩c(υ)=φ. We give here an alternative proof of the fact that every triangular-free induced subgraph of the triangular lattice is 5-[2]colourable. This proof yields a constant time distributed algorithm that finds a 5-[2]colouring of such a graph. We then give a distributed algorithm that finds a [p]colouring of a triangle-free induced subgraph of the triangular lattice with at most 5ωp(G)/4 + 3 colours.

FAST COMPUTATION OF THE WIENER INDEX OF FASCIAGRAPHS AND ROTAGRAPHS

The notion of a polygraph was introduced in chemical graph theory as a formalization of the chemical notion of polymers.’ Fasciagraphs and rotagraphs form an important class of polygraphs. In the language of graph theory they describe polymers with open ends and polymers that are closed upon themselves, respectively. They are highly structured, and this structure makes it possible to design efficient algorithms for computing several graph invariants.2 In this paper we show how the structure of fasciagraphs and rotagraphs can be used to obtain efficient algorithms for computing the Wiener index of such graphs. More precisely, if we regard basic arithmetic operations such as addition and multiplication to take a constant time, then the time complexity of our improved algorithms (theorem 5 ) depends only on the size k of a monograph in the polygraph and is independent of the number of monographs n. The paper is organized as follows. Motivation for studying such problems and definitions of polygraphs, rotagraphs, and fasciagraphs are given in section 1. Section 2 describes matrix approach to the computation of the Wiener index of fasciagraphs and rotagraphs. Two basic algorithms that realize this approach are presented (algorithms A and B). In section 3 possible extensions of these algorithms are briefly sketched. Using more sophisticated mathematical methods this approach is further extended, and the two algorithms

Deriving Formulas for Domination Numbers of Fasciagraphs and Rotagraphs

Recently, an algebraic approach which can be used to compute distance-based graph invariants on fasciagraphs and rotagraphs was given in [Mohar, Juvan, Žerovnik, Discrete Appl. Math. 80 (1997) 57-71]. Here we give an analogous method which can be employed for deriving formulas for the domination number of fasciagraphs and rotagraphs. In other words, it computes the domination numbers of these graphs in constant time, i.e. in time which depends only on the size and structure of a monograph and is independent of the number of monographs. Some further generalizations of the method are discussed, in particular the computation of the independent number and the k-coloring decision problem. Examples of fasciagraphs and rotagraphs include complete grid graphs. Grid graphs are one of the most frequently used model of processor interconnections in multiprocessor VLSI systems.

Unique square property and fundamental factorizations of graph bundles

Graph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167/168 (1998) 393) authors constructed an algorithm that finds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation δ* having the square property. An equivalence relation R on the edge set of a graph has the (unique) square property if and only if any pair of adjacent edges which belong to distinct R-equivalence classes span exactly one induced 4-cycle (with opposite edges in the same R-equivalence class). In this paper we define the unique square property and show that any weakly 2-convex equivalence relation possessing the unique square property determines the fundamental factorization of a graph as a nontrivial cartesian graph bundle over an arbitrary base graph, whenever it separates degenerate and nondegenerate edges of the factorization.

Recognizing weighted directed cartesian graph bundles

In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used. The first one is the new relation ~ ⁄ defined among the arcs of a digraph as a weighted directed analogue of the well-known relation ‐ ⁄ . The second one is the concept of half-convex subgraphs. A subgraph H is half-convex in G if any vertex x 2 G n H has at most one predecessor and at most one successor.

On domination numbers of graph bundles

Letγ(G) be the domination number of a graphG. It is shown that for anyκ ≥ 0 there exists a Cartesian graph bundleB█φF such thatγ(B█φF) =γ(B)γ(F) — 2κ. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequalityγ(B █ φF) ≤γ(B)γ(F) for strong graph bundles. Examples of graphsB andF withγ(B █ φF)γ(B)γ(F) are given.

An algorithm for K-convex closure and an application

An algorithm for computing the K-convex closure of a subgraph relative to a given equivalence relation R among edges of a graph is given.For general graph and arbitrary relation R the time complexity is O(q n 2 + mn), where n is the number of vertices, m is the number of edges and q is the number of equivalence classes of R.A special case is an O(mn) algorithm for the usual k-convexity.We also show that Cartesian graph bundles over triangle free bases can be recognized in O(mn) time and that all representations of such graphs as Cartesian graph bundles can be found in O(mn 2) time.

Recognizing Graph Products and Bundles

Problems of recognition of product graphs and graph bundles with respect to Cartesian, categorical, strong and lexicographic product are considered. A short survey of results and open problems is given.

Fault-diameter of generalized Cartesian products

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and D_c(G) denote the diameter of G after deleting any of its c \lt kG vertices. For a product of three factors G_1, G_2 and G_3, we prove that D_a+b+c+2(G) \lt D_a(G_1) + D_b(G_2) + D_c(G_3) + 1. We indicate how analogous proof gives the upper bound D_a+b+1(G) \lt D_a(G_1) + D_b(G_2) + 1 for the product of two factors. Finally, we show that D_a+b+1(G) \lt D_a(F) + D_b(B)+1 if G is a graph bundle with fibre F over base B, a \lt k_F,and b \lt k_B.