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- W2009475888 abstract "The balanced decomposition number $f(G)$ of a graph G was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339–3344]. A balanced coloring of a graph G is a coloring of some of the vertices of G with two colors, such that there is the same number of vertices in each color. Then, $f(G)$ is the minimum integer s with the following property: For any balanced coloring of G, there is a partition $V(G)=V_1,dotcup,cdots,dotcup,V_r$ such that, for every i, $V_i$ induces a connected subgraph, $|V_i|leq s$, and $V_i$ contains the same number of colored vertices in each color. Fujita and Nakamigawa studied the function $f(G)$ for many basic families of graphs, and demonstrated some applications. In this paper, we shall continue the study of the function $f(G)$. We give a characterization for noncomplete graphs G of order n which are $lfloorfrac{n}{2}rfloor$-connected, in view of the balanced decomposition number. We shall prove that a necessary and sufficient condition for such $lfloorfrac{n}{2}rfloor$-connected graphs G is $f(G)=3$. We shall also determine $f(G)$ when G is a complete multipartite graph, and when G is a generalized $Theta$-graph (i.e., a graph which is a subdivision of a multiple edge). Some applications will also be discussed. Further results about the balanced decomposition number also appear in two subsequent papers by Fujita and Liu." @default.
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- W2009475888 date "2010-01-01" @default.
- W2009475888 modified "2023-09-27" @default.
- W2009475888 title "The Balanced Decomposition Number and Vertex Connectivity" @default.
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- W2009475888 doi "https://doi.org/10.1137/090780894" @default.
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