Recognizing generalized Petersen graphs in linear time

Abstract

By identifying a local property which structurally classifies any edge, we show that the family of generalized Petersen graphs can be recognized in linear time.The generalized Petersen graphs, introduced by Coxeter [6] and named by Watkins [13], are cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. Various aspects of their structure have been extensively studied. Examples include identifying generalized Petersen graphs that are Hamiltonian [1, 2, 4], hypo-Hamiltonian [3], Cayley [10,12], or partial cubes [9], and finding their automorphism group [7] or determining isomorphic members of the family [11]. Additional aspects of the mentioned family are well surveyed in [5,8].

Fast recognition of some parametric graph families

“…For a fixed edge, counting the number of distinct 6 or 8-cycles it belongs to enabled us to design linear time recognition algorithms for I-graphs and double generalized Petersen graphs and a quasilinear one for folded hypercubes. To the best of our knowledge, in addition to this work, such a procedure was so far only used in [15] for the family of generalized Petersen graphs. We believe that a similar approach should give interesting results for other parametric graph families of bounded degree, such as Johnson graphs, rose window graphs, or Tabačjn graphs.…”

“…Additional aspects of the above-mentioned family are well surveyed in [6,11]. For this family, Watkins [25] had studied the structure of 8-cycles already in 1969, while linear recognition was settled by Krnc and Wilson [15].…”

“…He also described how the family of hypercubes is related to maximal [2,1,4]-cycle regular graphs and maximal [3,1,6]-cycle regular graphs. Generalized Petersen graphs are not [1, λ, 8]-cycle regular except in cases of G (3,1), G(4, 1), G (5,2), G (8,3), G (10,2), G (10,3), G (12,5), G (13,5), G (24,5) and G (26,5), which, as mentioned above, leads to a linear time recognition algorithm [15].…”

Under military war weapon support the economic bond level estimation using generalized Petersen graph with imputation

“…The generalized Petersen graph G(n,k) was introduced by Coxeter et al (1950) and named by Watkins (1969) from very interesting family of trivalent graphs that can be described by only two integer parameters. They include Hamiltonian and non-Hamiltonian graph, Bipartite and non-Bipartite graphs, vertex transitive and non-vertex transitive graphs, cayley and non-cayley graphs of girth 3,4,5,6,7 or 8 [Krnc, M. et al (2018)]. A generalized Petersen graph G(n,k) is a family of cubic graph who is 3-regular graph.…”

Some conditions implying stability of graphs

“…Canonical double covers have proven to play an important role in algebraic graph theory and have been studied by multiple groups of authors from a variety of perspectives [3, 11, 12, 15, 18–20, 24, 26]. It is well known that $$BX$$ is connected if and only if $$X$$ is connected and nonbipartite, see [7].…”