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The width of a node is the number of its parents, and the width of an ordered graph is the maximal width of its nodes. The induced graph of an ordered graph is obtained by adding some edges to an ordering graph, using the method outlined below. The induced width of an ordered graph is the width of its induced graph. [2] Given an ordered graph ...
Twin-width is defined for finite simple undirected graphs. These have a finite set of vertices, and a set of edges that are unordered pairs of vertices. The open neighborhood of any vertex is the set of other vertices that it is paired with in edges of the graph; the closed neighborhood is formed from the open neighborhood by including the vertex itself.
A cycle graph or circular graph of order n ≥ 3 is a graph in which the vertices can be listed in an order v 1, v 2, …, v n such that the edges are the {v i, v i+1} where i = 1, 2, …, n − 1, plus the edge {v n, v 1}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2.
A graph formed from a given graph by deleting one vertex, especially in the context of the reconstruction conjecture. See also deck, the multiset of all cards of a graph. carving width Carving width is a notion of graph width analogous to branchwidth, but using hierarchical clusterings of vertices instead of hierarchical clusterings of edges.
A graph G has treewidth k if and only if it has a haven of order k + 1 but of no higher order, where a haven of order k + 1 is a function β that maps each set X of at most k vertices in G into one of the connected components of G \ X and that obeys the monotonicity property that β(Y) ⊆ β(X) whenever X ⊆ Y.
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
In particular, for every infinite cardinal number κ there is an infinite partially ordered set of width ℵ 0 whose partition into the fewest chains has κ chains. [ 4 ] Perles (1963) discusses analogues of Dilworth's theorem in the infinite setting.
For arbitrary graph families, and arbitrary sentences, this problem is undecidable. However, satisfiability of MSO 2 sentences is decidable for the graphs of bounded treewidth, and satisfiability of MSO 1 sentences is decidable for graphs of bounded clique-width. The proof involves using Courcelle's theorem to build an automaton that can test ...