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The adaptation of the k -means algorithm to solving the multiple ellipses detection problem by using an initial approximation obtained by the DIRECT global optimization algorithm

Rudolf Scitovski, Kristian Sabo (2019)

Applications of Mathematics

We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse E is viewed as a Mahalanobis circle with center S , radius r , and some positive definite matrix Σ . A very efficient method for solving this problem is proposed. The method uses a modification of the k -means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers are determined...

The all-paths transit function of a graph

Manoj Changat, Sandi Klavžar, Henry Martyn Mulder (2001)

Czechoslovak Mathematical Journal

A transit function R on a set V is a function R V × V → 2 V satisfying the axioms u ∈ R ( u , v ) , R ( u , v ) = R ( v , u ) and R ( u , u ) = { u } , for all u , v ∈ V . The all-paths transit function of a connected graph is characterized by transit axioms.

The Balanced Decomposition Number of TK4 and Series-Parallel Graphs

Shinya Fujita, Henry Liu (2013)

Discussiones Mathematicae Graph Theory

A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, say red and blue, such that there is the same number of vertices in each colour. The balanced decomposition number f(G) of G is the minimum integer s with the following property: For any balanced colouring of G, there is a partition V (G) = V1 ∪˙ · · · ∪˙ Vr such that, for every i, Vi induces a connected subgraph of order at most s, and contains the same number of red and blue vertices. The function f(G)...

The basis number of some special non-planar graphs

Salar Y. Alsardary, Ali A. Ali (2003)

Czechoslovak Mathematical Journal

The basis number of a graph G was defined by Schmeichel to be the least integer h such that G has an h -fold basis for its cycle space. He proved that for m , n ≥ 5 , the basis number b ( K m , n ) of the complete bipartite graph K m , n is equal to 4 except for K 6 , 10 , K 5 , n and K 6 , n with n = 5 , 6 , 7 , 8 . We determine the basis number of some particular non-planar graphs such as K 5 , n and K 6 , n , n = 5 , 6 , 7 , 8 , and r -cages for r = 5 , 6 , 7 , 8 , and the Robertson graph.

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