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The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers...
We show that a graph G has no houses and no holes if and only if for every connected induced subgraph H of G and every vertex in H, either the vertex is adjacent to all the other vertices in H, or it forms a 2-pair of H with some other vertex in H. As a consequence, there is a simple linear time algorithm to find a 2-pair in HH-free graphs. We also note that the class of Meyniel graphs admits an analogous...
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