Geodetic Graphs Homeomorphic to a Given Geodetic Graph
📝 Abstract
This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry out a systematic finding of such a class of graphs. As a result, the enumeration of the class of geodetic graphs homeomorphic to certain Moore graphs has been performed.
💡 Analysis
This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry out a systematic finding of such a class of graphs. As a result, the enumeration of the class of geodetic graphs homeomorphic to certain Moore graphs has been performed.
📄 Content
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GEODETIC GRAPHS HOMEOMORPHIC TO A GIVEN GEODETIC GRAPH
Carlos E. Frasser, George N. Vostrov Department of Applied Mathematics, Odessa National Polytechnic University, Ukraine
This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry out a systematic finding of such a class of graphs. As a result, the enumeration of the class of geodetic graphs homeomorphic to certain Moore graphs has been performed.
Keywords: Geodetic System of Diophantine Equations, Geodetic Graph, Homeomorphism, Enumeration.
- Introduction.
If G is a graph, then V(G) and E(G) denote its vertex set and edge set, respectively. In this paper
a graph G is connected, undirected, and without loops or multiple edges. A path P from v0 to vn is
a sequence v0v1,…,vn of different successive vertices which are joined by an edge. A circuit C is
a sequence v0v1,…,vnv0 of different successive vertices which are also joined by an edge. The
length of P = v0v1,…,vn is the number of edges it contains and will be denoted either by P =
v0v1,…,vn or by e1e2,…,en, where e1, e2,…,en is a sequence of edges that join vertices
v0 ,v1,…,vn. A circuit C is even or odd if its length is even or odd, respectively. We can assign an
orientation to each circuit C. If C is an even circuit and u, v ϵ V(C), we say that u, v are
C-opposite if C(u, v) = C(v, u).
The distance between u, v ϵ V(G), denoted dG(u, v), is the length of a shortest path connecting
these vertices. The diameter of G, denoted d(G), is the greatest distance between any pair of its
vertices. If u and v are two different vertices of G, a geodesic (u, v) in G, is a path of shortest
length whose endpoints are u and v. Clearly, dG(u, v) = (u, v). A Graph G is geodetic [6] if,
given any vertices u and v of G, there exists a unique geodesic (u, v) in G.
The degree of a vertex v, denoted deg(v), is the number of edges incident to v. G is said to be
regular of degree k if every vertex of G has the same degree k. A node v is defined as a vertex
for which deg(v) ≥ 3. A segment is a path whose only nodes are its endpoints u and v and it is
denoted S(u, v). The girth of a graph G, denoted g(G), is the length of a shortest circuit contained
in G.
Two graphs G1 and G2 are said to be isomorphic, denoted G1 G2, if there exists a one-to-one
function from V(G1) onto V(G2) that preserves adjacency. Two graphs are said to be homeo-
morphic if and only if each can be obtained from the same graph by insertion of vertices onto the
edges of one or both graphs. Notice that every graph is homeomorphic to itself. A complete
graph is a graph in which every pair of vertices is adjacent. The complete graph with n vertices is
denoted Kn. A block is a graph with vertex connectivity > 1.
If H is a subgraph of G, G – H denotes the subgraph of G obtained by deleting V(H) from
V(G) and removing all edges from G that have an endpoint in V(H). A subset S of V(G) is said to
generate a subgraph if H is the section subgraph on S; that is, V(H) = S and H contains all edges
of G connecting two vertices of S [11]. A clique is defined as a maximal complete subgraph Kn,
n ≥ 3; that is, a complete subgraph on at least three vertices which is contained in no larger com-
plete subgraph.
2
A regular graph G of degree k and diameter d is called a Moore graph if
d
V(G) = 1 + k (k - 1) i - 1 i = 1 For a Moore graph of diameter 2, this condition becomes V(G) = 1 + k2. From Stemple [11] and Hoffman & Singleton [4], we have
Theorem 1. Let G be a geodetic block of diameter 2. If G does not contain cliques Kn, n 3, then G is a Moore graph, where k = 2, 3, 7, 57. If k = 2, then G is a circuit of length 5. If k = 3, then G is the Petersen graph. If k = 7, then G is the Hoffman-Singleton graph.
The existence of a graph as described in the previous theorem with k = 57 is still undecided.
Now, we will examine some results related to geodetic graphs which are formulated in terms
of the general ideas previously exposed. The following characterization was established by Stemple and Watkins [13].
Lemma 1. A connected graph G is geodetic if and only if G does not contain an even circuit C such that for every pair of C-opposite vertices u, v dG(u, v) = d(C).
From Stemple [12], we have
Theorem 2. Let G be homeomorphic to K4; G is geodetic if and only if: (1) The six segments of G are geodesics. (2) Each circuit of G that contains exactly three segments is odd. (3) All circuits of G that contain exactly four segments have equal length.
A graph G homeomorphic to K4 which satisfies Theorem 2’s conditions (1), (2), and (3) is
shown in Figure 1.
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