Characterizing and Recognizing Probe Block Graphs

Graph Theory International audience Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

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Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

Algebra Colloquium ◽

10.1142/s1005386708000370 ◽

2008 ◽

Vol 15 (03) ◽

pp. 379-390 ◽

Cited By ~ 1

Author(s):

Xuesong Ma ◽

Ruji Wang

Keyword(s):

Automorphism Group ◽

Bipartite Graphs ◽

Type Ii ◽

Symmetric Graph ◽

Trivalent Graph ◽

Block Graphs ◽

Group A ◽

Vertex Transitive ◽

Vertex Transitive Graphs ◽

Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

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Graphical splittings of Artin kernels

Journal of Group Theory ◽

10.1515/jgth-2020-0124 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Enrique Miguel Barquinero ◽

Lorenzo Ruffoni ◽

Kaidi Ye

Keyword(s):

Free Group ◽

Artin Group ◽

Artin Groups ◽

Graphs Of Groups ◽

Underlying Graph ◽

Block Graphs ◽

Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

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