The Degree Realization problem requires, given a sequence $d$ of $n$ positive integers, to decide whether there exists a graph whose degrees correspond to $d$, and to construct such a graph if it exists. A more challenging variant of the problem arises when $d$ has many different realizations, and some of them may be more desirable than others. We study \emph{optimized realization} problems in which the goal is to compute a realization that optimizes some quality measure. Efficient algorithms are known for the problems of finding a realization with the maximum clique, the maximum independent set, or the minimum vertex cover. In this paper, we focus on two problems for which such algorithms were not known. The first is the Degree Realization with Minimum Dominating Set problem, where the goal is to find a realization whose minimum dominating set is minimized among all the realizations of the given sequence $d$. The second is the Degree Realization with Maximum Matching problem, where the goal is to find a realization with the largest matching among all the realizations of $d$. We present polynomial time realization algorithms for these two open problems. A related problem of interest and importance is \emph{characterizing} the sequences with a given value of the optimized function. This leads to an efficient computation of the optimized value without providing the realization that achieves that value. For the Maximum Matching problem, a succinct characterization of degree sequences with a maximum matching of a given size was known. This paper provides a succinct characterization of sequences with minimum dominating set of a given size.

翻译：度实现问题要求在给定一个包含n个正整数的序列d时，判断是否存在一个图，其度序列与d相匹配，并在存在时构造出这样的图。当d存在多种不同的实现方式，且某些实现可能比其他实现更理想时，问题会变得更加复杂。我们研究的是优化实现问题，其目标是计算能够优化某些质量度量的实现。对于寻找具有最大团、最大独立集或最小顶点覆盖的实现问题，已有高效的算法。本文重点关注两个尚未有已知算法的此类问题。第一个是最小支配集度实现问题，其目标是在给定序列d的所有实现中，找到一个最小支配集最小的实现。第二个是最大匹配度实现问题，其目标是在d的所有实现中，找到一个具有最大匹配的实现。我们针对这两个开放问题提出了多项式时间的实现算法。一个相关且重要的问题是对具有给定优化函数值的序列进行特征刻画。这可以在不提供实现该值的具体实现的情况下，高效地计算出优化值。对于最大匹配问题，已知具有给定大小最大匹配的度序列存在简洁的特征刻画。本文则提供了具有给定大小最小支配集的序列的简洁特征刻画。