用指定度和邻度序列构造有界度图
Constructing bounded degree graphs with prescribed degree and neighbor degree sequences
摘要：
设$D = d_1, d_2, \ldots, d_n$和$F = f_1, f_2,\ldots, f_n$是两个正整数序列。我们考虑以下决策问题：是否存在$i)$多重图，$ii)$无环多重图，$iii)$简单图，$iv)$连通简单图，$v)$树，$vi)$毛虫$G = (V,E)$使得对所有$k$，$d(v_k) = d_k$和$\sum_{w\in \mathcal{N}(v_k)} d(w) = f_k$（$d(v)$是$v$的度，$\mathcal{N}(v)$是$v$的邻居集）.在这里，我们表明，所有这些决策问题可以在多项式时间内解决，如果$\max_{k} d_k$是有界的。这个问题是由碳氢化合物的NMR光谱激发的。
英文摘要：
Let $D = d_1, d_2, \ldots, d_n$ and $F = f_1, f_2,\ldots, f_n$ 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 $\sum_{w\in \mathcal{N}(v_k)} d(w) = f_k$ ($d(v)$ is the degree of $v$ and $\mathcal{N}(v)$ 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.
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