Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

09/27/2021

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by   Uroš Čibej, et al.

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Let D = d_1, d_2, …, d_n and F = f_1, f_2,…, f_n be two sequences of positive integers. We consider the following decision problems: is there a i) multigraph, ii) loopless multigraph, iii) simple graph, iv) connected simple graph, v) tree, vi) caterpillar G = (V,E) such that for all k, d(v_k) = d_k and ∑_w∈𝒩(v_k) d(w) = f_k (d(v) is the degree of v and 𝒩(v) is the set of neighbors of v). Here we show that all these decision problems can be solved in polynomial time if max_k d_k is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.