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Vertex-colored encompassing graphs. (English) Zbl 1298.05120

Summary: It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.

MSC:

05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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