Linear-time certifying algorithms for near-graphical sequences

Abstract

A sequence 〈 d 1 , d 2 , … , d n 〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose i th vertex has degree d i , for 1 ≤ i ≤ n . The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs. If H = ( V , E ) is a graph and g and f are integer-valued functions on the vertex set V , then a ( g , f ) -factor of H is a subgraph G = ( V , F ) of H whose degree at each vertex v ∈ V lies in the interval [ g ( v ) , f ( v ) ] . Thus, a ( 0 , 1 ) -factor is just a matching of H and a (1, 1)-factor is a perfect matching of H . If H is complete then a ( g , f ) -factor realizes a degree sequence that is consistent with the sequence of intervals 〈 [ g ( v 1 ) , f ( v 1 ) ] , [ g ( v 2 ) , f ( v 2 ) ] , … , [ g ( v n ) , f ( v n ) ] 〉 . Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient ( g , f ) -factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient ( g , f ) -factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing ( f , f ) -factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös–Gallai characterization of ( f , f ) -factors in complete graphs) at no additional cost.