A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the graph. In this work we study the problem of finding a minimal connected tropical subgraph. We first show that this problem is NP-Hard for trees, interval graphs and split graphs, but polynomial when the number of colors is logarithmic in terms of the order of the graph (i.e. FPT). We then provide upper bounds for the order of the minimal connected tropical subgraph under various conditions. We finally study the problem of finding a connected tropical subgraph in a randomly vertex-colored random graph.

Source : oai:HAL:hal-01352845v1

Volume: Vol. 17 no. 3

Section: Graph Theory

Published on: August 7, 2016

Submitted on: February 2, 2015

Keywords: vertex-colored random graph,vertex-colored graph, connected subgraph, tropical subgraph, colorful subgraph,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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