Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments

Playing with Derivation Modes and Halting Conditions

In the area of P systems, besides the standard maximally parallel derivation mode, many other derivation modes have been investigated, too. In this paper, many variants of hierarchical P systems and tissue P systems using different derivation modes are considered and the effects of using di erent derivation modes, especially the maximally parallel derivation modes and the maximally parallel set derivation modes, on the generative and accepting power are illustrated. Moreover, an overview on some control mechanisms used for (tissue) P systems is given. Furthermore, besides the standard total halting mode, we also consider different halting conditions such as unconditional halting and partial halting and explain how the use of different halting modes may considerably change the computing power of P systems and tissue P systems

All graphs have tree-decompositions displaying their topological ends

We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992

Permanents, Pfaffian orientations, and even directed circuits

Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all of the above problems are equivalent. We prove a structural characterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic nonplanar bipartite graph.Comment: 47 pages, published versio

Extremal \u3cem\u3eH\u3c/em\u3e-Colorings of Trees and 2-connected Graphs

For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H -colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n -vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs