A path-neighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3-sun-free path-neighborhood graphs are characterized. The 3-sun is obtained from a 6-cycle by adding three chords between the three pairs of vertices at distance 2. A Pk-graph is a path-neighborhood graph in which every neighborhood is a Pk, where Pk is the path on k vertices. The Pk-graphs are characterized for k ≤ 4.
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 as well as the set of leaps of the connected graph G. The underlying graph of , as well...
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...
The well known types of routes in graphs and directed graphs, such as walks, trails, paths, and induced paths, are characterized using axioms on vertex sequences. Thus non-graphic characterizations of the various types of routes are obtained.
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.
In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.
The main subject of our study are spherical (weakly spherical) graphs, i.e. connected graphs fulfilling the condition that in each interval to each vertex there is exactly one (at least one, respectively) antipodal vertex. Our analysis concerns properties of these graphs especially in connection with convexity and also with hypercube graphs. We deal e.g. with the problem under what conditions all intervals of a spherical graph induce hypercubes and find a new characterization of hypercubes: is...
A transit function on a set is a function satisfying the axioms , and , for all . The all-paths transit function of a connected graph is characterized by transit axioms.
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