Abstract
We consider the problem of realizable interval sequences. An interval sequence is comprised of $n$ integer intervals $[a_i,b_i]$ such that $0\le a_i\leq b_i \le n-1$ and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n),$ satisfying the condition $a_i\leq d_i\leq b_i$ for each $i\in[1,n]$. There is a characterization (also implying an $O(n)$ verifying algorithm) known for realizability of interval sequences, which is a generalization of the Erdös--Gallai characterization for graphic sequences. However, given any realizable interval sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is nonrealizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence in the same time. Finally, we consider variants of the problem, such as computing the most-regular graphic sequence and computing a minimum extension of a length $p$ nongraphic sequence to a graphic one.
Highlights
The Graph Realization problem for a property P deals with the following existential question: Does there exist a graph that satisfies the property P ? Its fundamental importance is apparent, ranging from better theoretical understanding, to network design questions
We provide an O(n log n) time algorithm for computing a graphic sequence for any realizable interval sequence
Our algorithms are based on a novel divide and conquer methodology, wherein we show that partitioning a realizable interval sequence along any levelled sequence guarantees that at least one of the new child interval sequences is realizable
Summary
The Graph Realization problem for a property P deals with the following existential question: Does there exist a graph that satisfies the property P ? Its fundamental importance is apparent, ranging from better theoretical understanding, to network design questions (such as constructing networks with certain desirable connectivity properties). Cai et al [5] extended Erdös and Gallai’s work by providing an easy to verify characterization for realizable interval-sequences, thereby resolving Question 1 Their result crucially uses the (g, f )-Factor Theorem of Lovász [23]. Our new approach enables us to tackle an optimization version of the problem in which it is required to compute the “most regular” sequence realizing the given interval sequence S, using the natural measure of the minimum sum of pairwise degree differences, i,j |di − dj|, as our regularity measure. There exists an algorithm that for any integer n ≥ 1 and any length n realizable interval sequence S, computes the most regular graphic sequence realizing interval sequence S (i.e., the one minimizing the sum of pairwise degree difference), in time O(n2).
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