An L(2, 1)-labelling of a graph G is an assignment of non-negative integers to the vertices of G such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The L(2, 1)-labelling number of G, denoted by λ(G), is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient ...

We present a fast, interactive method for separating bones that have been collectively segmented from a CT volume. Given user-provided seed points, the method computes the separation as a multi-way cut on a weighted graph constructed from the binary, segmented volume. By properly designing and weighting the graph, we show that the resulting cut can accurately be placed at bone-interfaces using ...

Given an integer r 0, let G r = (V r ; E) denote a graph consisting of a simple nite undirected graph G = (V; E) of order n and size m together with r isolated vertices K r. Then jV j = n, jV r j = n + r, and jEj = m. Let L : V r ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every d...

The Lovász θ function of a graph, a fundamental tool in combinatorial optimization and approximation algorithms, is computed by solving a SDP. In this paper we establish that the Lovász θ function is equivalent to a kernel learning problem related to one class SVM. This interesting connection opens up many opportunities bridging graph theoretic algorithms and machine learning. We show that ther...

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we prove that any balanced bipartite graph with minimum degree greater than |V (G)|/4 ≥ 2 is magic. A similar result is presented for supermag...

Motivated by the conjecture on the L(2, 1)-labelling number λ(G) of a graph G by Griggs and Yeh [2] and the question: “Is the upper bound (∆+3)/4 for λ(G) for chordal graphs with maximum degree ∆ is sharp?”, posed by Sakai [3], we study the bounds for λ(G) for chordal graphs in this paper. Let G be a chordal graph on n vertices with maximum degree ∆ and maximum clique number ω. We improve the u...

a. To check that the graph is represented correctly, Bob asks Alice to remove all the paper cups covering the endpoints of every edge. This reveals an isomorphism to the original graph, which Bob cannot check efficiently; thus, Bob also removes the paper cups corresponding to the green numbers, revealing the original labelling of the vertices. This reveals nothing about the vertex cover, but al...

We present a graph partitioning method to integrate prior knowledge in data grouping. We consider priors represented by three types of constraints: unitary constraints on labelling of groups, partial a priori grouping information, external in uence on binary constraints. They are modelled as biases in the grouping process. We incorporate these biases into graph partitioning criteria. Computatio...

Object recognition using graph-matching techniques can be viewed as a two-stage process: extracting suitable object primitives from an image and corresponding models, and matching graphs constructed from these two sets of object primitives. In this paper we concentrate mainly on the latter issue of graph matching, for which we derive a technique based on probabilistic relaxation graph labelling...

The paper describes a method for the analysis of the content of a binary image in order to find its structure. The class of images it deals with consists of images showing a groups of objects connected one to another forming a graph-like structure. Proposed method extracts automatically this structure from image bitmap and produces graph adjacency matrix describing it. The method is based on mo...