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A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering. [3] Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that a perfect ...
In particular, the same lexicographic breadth-first search algorithm used to find perfect elimination orders of chordal graphs can be used to find semiperfect elimination orders of distance-hereditary graphs, which are therefore also perfectly orderable. [6] The graphs for which every vertex ordering is a perfect ordering are the cographs ...
Least fixed point based logics of graphs extend the first-order logic of graphs by allowing predicates (properties of vertices or tuples of vertices) defined by special fixed-point operators. This kind of definition begins with an implication, a formula stating that when certain values of the predicate are true, then other values are true as well.
First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate.
The algorithm is called lexicographic breadth-first search because the order it produces is an ordering that could also have been produced by a breadth-first search, and because if the ordering is used to index the rows and columns of an adjacency matrix of a graph then the algorithm sorts the rows and columns into lexicographical order.
Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields. [1] It can be used for inference of maximum a posteriori (MAP) state or estimation of conditional or marginal distributions over a subset of variables.
In first-order logic, resolution condenses the traditional syllogisms of logical inference down to a single rule. To understand how resolution works, consider the following example syllogism of term logic: All Greeks are Europeans. Homer is a Greek. Therefore, Homer is a European. Or, more generally: .