146 research outputs found
Independent sets and non-augmentable paths in generalizations of tournaments
AbstractWe study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction
Notes on Schubert, Grothendieck and Key Polynomials
We introduce common generalization of (double) Schubert, Grothendieck,
Demazure, dual and stable Grothendieck polynomials, and Di
Francesco-Zinn-Justin polynomials. Our approach is based on the study of
algebraic and combinatorial properties of the reduced rectangular plactic
algebra and associated Cauchy kernels
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Integral matrices with given row and column sums
AbstractLet P = (pij) and Q = (qij) be m × n integral matrices, R and S be integral vectors. Let UPQ(R, S) denote the class of all m × n integral matrices A with row sum vector R and column sum vector S satisfying P ⩽ A ⩽ Q. For a wide variety of classes UPQ(R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in UPQ(R, S). The first characterization unifies the results of Gale-Ryser, Fulkerson, and Anstee. Many other properties of (0, 1)-matrices with prescribed row and column sum vectors generalize to integral classes satisfying the main condition. We also study the decomposibility of integral classes satisfying the main condition. As a consequence of our decomposibility theorem, it follows a theorem of Beineke and Harary on the existence of a strongly connected digraph with given indegree and outdegree sequences. Finally, we introduce the incidence graph for a matrix in an integral class UPQ(R, S) and study the invariance of an element in a matrix in terms of its incidence graph. Analogous to Minty's Lemma for arc colorings of a digraph, we give a very simple labeling algorithm to determine if an element in a matrix is invariant. By observing the relationship between invariant positions of a matrix and the strong connectedness of its incidence graph, we present a very short graph theoretic proof of a theorem of Brualdi and Ross on invariant sets of (0, 1)-matrices. Our proof also implies an analogous theorem for a class of tournament matrices with given row sum vector, as conjectured by the analogy between bipartite tournaments and ordinary tournaments
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
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