Abstract
Abstract The graph sandwich problem for property Π is defined as follows: Given two graphs G 1=(V,E 1) and G 2=(V,E 2) such that E 1⊆E 2, is there a graph G=(V,E) such that E 1⊆E⊆E 2 which satisfies property Π? We propose to study sandwich problems for properties Π concerning orientations, such as Eulerian orientation of a mixed graph and orientation with given in-degrees of a graph. We present a characterization and a polynomial-time algorithm for solving the m-orientation sandwich problem.
Highlights
Given two graphs G1 = (V, E1) and G2 = (V, E2) with the same vertex set V and E1 ⊆ E2, a graph G = (V, E) is called a sandwich graph for the pair G1, G2 if E1 ⊆ E ⊆ E2
We call E1 the mandatory edge set, E0 = E2 \ E1 the optional edge set and E3 the forbidden edge set, where E3 denotes the set of edges of the complementary graph G2 of G2
Graph sandwich problems have attracted much attention lately arising from many applications and as a natural generalization of recognition problems [1,2,3, 7, 22, 24]
Summary
The following theorem characterizes graphs having an orientation with a given in-degree vector. Theorem 3 (Hakimi [13]) Given an undirected graph G = (V , E) and a non-negative integer vector m on V , there exists an orientation G of G whose in-degree vector is m if and only if m(X) ≥ i(X) ∀X ⊆ V ,. Given an undirected graph G = (V , E) and a nonnegative integer vector m on V , let m G = mbe the set function defined on E by m (F ) = m(V (F )) where V (F ) is the set of vertices covered by F. Claim 2 The set {F ⊆ E : m(X) ≥ iF (X), ∀X ⊆ V } is the set of independent sets of the matroid Mm
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