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For example, in the simple equation 3 + 2y = 8y, both sides actually contain 2y (because 8y is the same as 2y + 6y). Therefore, the 2y on both sides can be cancelled out, leaving 3 = 6y, or y = 0.5. This is equivalent to subtracting 2y from both sides. At times, cancelling out can introduce limited changes or extra solutions to an equation. For ...
The following is a well-known result characterizing the minimal separators: [3] Lemma. A vertex separator S in G is minimal if and only if the graph G – S, obtained by removing S from G, has two connected components C 1 and C 2 such that each vertex in S is both adjacent to some vertex in C 1 and to some vertex in C 2.
A graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of ΔY- or YΔ-transformations and the following normalization steps: removing a loop, removing a parallel edge, removing a vertex of degree one, smoothing out a vertex of degree two, i.e., replacing it by a single edge between its two former neighbors.
The dual graph of a line arrangement has one node per cell and one edge linking any pair of cells that share an edge of the arrangement. These graphs are partial cubes, graphs in which the nodes can be labeled by bitvectors in such a way that the graph distance equals the Hamming distance between labels.
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...
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Thus, a split can be described as a partition of the vertices of the graph into two subsets X and Y, such that every neighbor of X in Y is adjacent to every neighbor of Y in X. [2] A cut or split is trivial when one of its two sides has only one vertex in it; every trivial cut is a split. A graph is said to be prime (with respect to splits) if ...
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