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Generalizing graph decompositions
2021The Latin aphorism ‘divide et impera’ conveys a simple, but central idea in mathematics and computer science: ‘split your problem recursively into smaller parts, attack the parts, and conquer the whole’. There is a vast literature on how to do this on graphs.
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Decomposition of Directed Graphs
SIAM Journal on Algebraic Discrete Methods, 1982A composition for directed graphs which generalizes the substitution (or X-join) composition of graphs and digraphs, as well as the graph version of set-family composition, is described. It is proved that a general decomposition theory can be applied to the resulting digraph decomposition.
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String decompositions of graphs
Ars Comb., 1998The authors prove, among other results, that a 2-connected graph \(G=(V,E)\) has a string decomposition (SD) iff \(G\) is not a cycle. An SD is a partition \(P\) of \(E\) into strings, which are defined to be the walks in \(G\) in which inner vertices do not repeat and the two distinct endvertices repeat each at most twice (a cycle is not a string ...
Atsushi Kaneko, Mamoru Watanabe
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Decompositions of Complete Graphs
Bulletin of the London Mathematical Society, 2000Summary: If \(s_1,s_2,\dots, s_t\) are integers such that \(n-1= s_1+ s_2+\cdots+ s_t\) and such that for each \(i\) \((1\leq i\leq t)\), \(2\leq s_i\leq n-1\) and \(s_in\) is even, then \(K_n\) can be expressed as the union \(G_1\cup G_2\cup\cdots\cup G_t\) of \(t\) edge-disjoint factors, where for each \(i\), \(G_i\) is \(s_i\)-connected.
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P4-decompositions of regular graphs
Journal of Graph Theory, 1999It is shown that every simple \(r\)-regular graph \(G\) admits a balanced \(P_4\)-decomposition if \(r \equiv 0\pmod 3\) and \(G\) has no cut-edge when \(r\) is odd. It is also shown that a connected 4-regular graph \(G\) admits a \(P_4\)-decomposition if and only if \(| E(G)| \equiv 0\pmod 3\) by characterizing graphs of maximum degree 4 that admit a ...
Katherine Heinrich +2 more
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A Decomposition Dynamic graph convolutional recurrent network for traffic forecasting
Pattern Recognition, 2023Wenchao Weng +6 more
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On the decomposition of graphs into cliques
Ars Comb., 2000A clique decomposition of a graph \(G\) is called greedy if it is obtained by successively removing maximal cliques from \(G\). The paper shows that the sum of the orders of the cliques in such a decomposition is less than \(\frac {5}{8}n^2\).
Gregory F. Bachelis +2 more
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Random Structures and Algorithms, 1998
Summary: Let \(H\) be a tree on \(h\geq 2\) vertices. It is shown that if \(G=(V,E)\) is a graph with \(\delta(G)\geq(| V|/2) +10h^4 \sqrt{| V| \log| V|}\), and \(h-1\) divides \(| E|\), then there is a decomposition of the edges of \(G\) into copies of \(H\). This result is asymptotically the best possible for all trees with at least three vertices.
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Summary: Let \(H\) be a tree on \(h\geq 2\) vertices. It is shown that if \(G=(V,E)\) is a graph with \(\delta(G)\geq(| V|/2) +10h^4 \sqrt{| V| \log| V|}\), and \(h-1\) divides \(| E|\), then there is a decomposition of the edges of \(G\) into copies of \(H\). This result is asymptotically the best possible for all trees with at least three vertices.
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On the homogeneous decomposition of graphs
1993We introduce and investigate the notion of p-connectedness. As it turns out, this concepts leads naturally to a unique tree representation for arbitrary graphs: the leaves of this tree are the p-connected components along with weak vertices, that is, vertices of the graph that belong to no p-connected component. By refining this first result, we obtain
Beverly Jamison, Stephan Olariu
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Atoll decompositions of graphs
Journal of Graph Theory, 1982AbstractAn island decomposition of a graph G consists of a set of vertex‐disjoint paths which cover the vertex set of G. If the endpoints of the paths are mutually nonadjacent, then we have an atoll decomposition. We characterize graphs requiring two paths in an island decomposition yet having no atoll decomposition.
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