
From Modular Decomposition Trees to Rooted Median Graphs
The modular decomposition of a symmetric map δ X× X →Υ (or, equivalently...
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All longest cycles intersect in partial 3trees
We show that all longest cycles intersect in 2connected partial 3trees...
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On bounds on bend number of classes of split and cocomparability graphs
A kbend path is a rectilinear curve made up of k + 1 line segments. A B...
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B1EPG representations using blockcutpoint trees
In this paper, we are interested in the edge intersection graphs of path...
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Decomposition of (2k+1)regular graphs containing special spanning 2kregular Cayley graphs into paths of length 2k+1
A P_ℓdecomposition of a graph G is a set of paths with ℓ edges in G tha...
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On bounds on bend number of split and cocomparability graphs
A path is a simple, piecewise linear curve made up of alternating horizo...
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Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1Sperner hypergraphs
A hypergraph H is said to be 1Sperner if for every two hyperedges the s...
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SplitDecomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs
In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the splitdecomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphsin particular the enumeration of several subclasses of perfect graphs (distancehereditary, 3leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the splitdecomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper, can be thought of as trees with their edges replaced by cycles (of arbitrary lengths). Their splitdecomposition trees contain prime nodes that are cycles, making them ideal to study. We derive a characterization for the splitdecomposition trees of cactus graphs, produce a general template of symbolic grammars for cactus graphs, and implement random generation for these graphs, building on work by Iriza (2015).
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