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A balanced coloring of a graph $$G$$ G means a triple $$\{P_1,P_2,X\}$$ { P 1 , P 2 , X } of mutually disjoint subsets of the vertex-set $$V(G)$$ V ( G ) such that $$V(G)=P_1 \dot{\cup } P_2 \dot{\cup } X$$ V ( G ) = P 1 ∪ ˙ P 2 ∪ ˙ X and $$|P_1|=|P_2|$$ | P 1 | = | P 2 | . A balanced decomposition associated with the balanced...
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree-depth. In this paper we survey results related to this concept and particularly explain how these graphs are used to decompose and construct more complex graphs and structures. In more technical terms we survey some of the properties...
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