Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidal-triple-free graphs, permutation graphs, and co-comparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of M"uller. 1 Background A graph H is an interval graph if it is the intersection graph of a family of intervals Iv, v 2 V (H). (Two vertices v; v 0 are adjacent in H if and only if Iv and Iv0 intersect.) If the

For graphs G and H, a mapping f: V (G)→V (H) is a homomorphism of G to H if uv ∈ E(G) implies f(u)f(v) ∈ E(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is � u∈V (G) cf(u)(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs ci(u), u ∈ V (G), i ∈ V (H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NP-hard. This solves an open problem from an earlier paper. 1

For digraphs D and H, a mapping f: V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H, the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM(H). An optimization version of the homomorphism problem was motivated by a realworld problem in defence logistics and was introduced in [13]. If each vertex u ∈ V (D) is associated with costs ci(u), i ∈ V (H), then the cost of the homomorphism f is u∈V (D) c f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H and denote it as MinHOM(H). The problem is to decide, for an input graph D with costs ci(u), u ∈ V (D), i ∈ V (H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM(H) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM(H) were proved when H is a semicomplete digraph [10], and a semicomplete multipartite digraph [12, 11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in [9].

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Abstract. We give two new algorithms for recognizing proper circulararc graphs and unit circular-arc graphs. The algorithms either provide a model for the input graph, or a certificate that proves that such a model does not exist and can be authenticated in O(n) time. 1

A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easy-to-verify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manually or by use of a program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust the algorithm. We put forward the thesis that certifying algorithms are much superior to non-certifying algorithms, and that for complex algorithmic tasks, only certifying algorithms are satisfactory. Acceptance of this thesis would lead to a change of how algorithms are taught and how algorithms are researched. The widespread use of certifying algorithms would greatly enhance the reliability of algorithmic software. We survey the state of the art in certifying algorithms and add to it. In particular, we start a

This is not a survey article. Rather it is a personal statement written for a lifelong friend and collaborator. Still it is an ambition of this article to trace some of the key moments of our development in the past 40 years. In doing so perhaps some evidence has arisen which otherwise seems to be obscured by the hectic day-to-day academic life. Thus the title.