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Modular and median signpost systems and their underlying graphs

100%
Mulder H., Nebeský L.
Discussiones Mathematicae Graph Theory
|
2003
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tom 23
|
nr 2
309-324
EN
The concept of a signpost system on a set is introduced. It is a ternary relation on the set satisfying three fairly natural axioms. Its underlying graph is introduced. When the underlying graph is disconnected some unexpected things may happen. The main focus are signpost systems satisfying some extra axioms. Their underlying graphs have lots of structure: the components are modular graphs or median graphs. Yet another axiom guarantees that the underlying graph is also connected. The main results of this paper concern if-and-only-if characterizations involving signpost systems satisfying additional axioms on the one hand and modular, respectively median graphs on the other hand.
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Leaps: an approach to the block structure of a graph

100%
Mulder H., Nebeský L.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 1
77-90
EN
To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation $+_G$ as well as the set of leaps $L_G$ of the connected graph G. The underlying graph of $+_G$, as well as that of $L_G$, turns out to be just the block closure of G (i.e., the graph obtained by making each block of G into a complete subgraph).
3
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On 3-simplicial vertices in planar graphs

81%
Boros E., Jamison R., Laskar R., Mulder H.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
|
nr 3
413-421
EN
A vertex v in a graph G = (V,E) is k-simplicial if the neighborhood N(v) of v can be vertex-covered by k or fewer complete graphs. The main result of the paper states that a planar graph of order at least four has at least four 3-simplicial vertices of degree at most five. This result is a strengthening of the classical corollary of Euler's Formula that a planar graph of order at least four contains at least four vertices of degree at most five.
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