A Characterization of Geodetic Graphs in Terms of their Embedded Even Graphs

Abstract

The problem of finding the general classification of geodetic graphs is still open. We believe that one of the obstacles to attain this goal is that geodetic graphs lack a structural description. In other words, their fundamental properties have not yet been established in terms of the description of the complete graphs, paths and cycles contained within them. The absence of this information makes their generation and enumeration (as inherent parts of their general classification) a difficult task. This paper examines the structural qualities of geodetic graphs using their so-called embedded even graphs. To this effect, the necessary and sufficient conditions for eliminating the nongeodecity of each pair of C-opposite vertices in an even cycle C have been formulated, while the bigeodecity of the embedded even graphs of a geodetic graph has been established. In a sense, this allows us to arrive at the conclusion that the basic building blocks of geodetic graphs are precisely this class of bigeodetic ones.