27 research outputs found

    On the sparsity order of a graph and its deficiency in chordality

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    Given a graph GG on nn nodes, let {cal P_G denote the cone consisting of the positive semidefinite ntimesnntimes n matrices (with real or complex entries) having a zero entry at every position corresponding to a non edge of GG. Then, the order of GG is defined as the maximum rank of a matrix lying on an extreme ray of the cone {cal P_G. It is shown in [AHMR88] that the graphs of order 1 are precisely the chordal graphs and a characterization of the graphs having order 22 is conjectured there in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with order 2 in the complex case and we give a decomposition result for the graphs having order le2le 2 in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also establish an inequality relating the order {rm ord_{oF(G) of a graph GG (oF=oRoF=oR or oCoC) and the parameter {rm fill(G) defined as the minimum number of edges needed to be added to GG in order to obtain a chordal graph. Namely, we show that {rm ord_{oF(G)le 1 +epsilon_oF cdot {rm fill(G) where epsilonoR=1epsilon_oR=1 and epsilonoC=2epsilon _oC =2; this settles a conjecture posed in [HPR89]

    Maximal chordal subgraphs

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    AbstractAn algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations

    Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

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    Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
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