Abstract This study grew from an attempt to give a local analysis of matroid base graphs. A neighborhood-preserving covering of graphs p:G → H is one such that p restricted to every neighborhood in G is an isomorphism. This concept arises naturally when considering graphs with a prescribed set of local properties. A characterization is given of all connected graphs with two local properties: (a) there is a pair of adjacent points, the intersection of whose neighborhoods does not contain three mutually nonadjacent points; (b) the intersection of the neigh-borhoods of points two apart is a 4-cycle. Such graphs have neighborhoods of the form Kn × Km for fixed n, m and are either complete matroid base graphs or are their images under neighborhood-preserving coverings. If n ≠ m, the graph is unique; if n = m, there are n − 3 such images which are nontrivial. These examples prove that no set of properties of bounded diameter can characterize matroid base graphs.
[1]
A. Gewirtz,et al.
Graphs with Maximal Even Girth
,
1969,
Canadian Journal of Mathematics.
[2]
F. Harary,et al.
On the Tree Graph of a Matroid
,
1972
.
[3]
W. Mayeda,et al.
On the Realizability of a Set of Trees
,
1961
.
[4]
S. Hakimi.
On Realizability of a Set of Trees
,
1961
.
[5]
R. C. Bose,et al.
A Characterization of Tetrahedral Graphs
,
1967
.
[6]
H. Whitney.
On the Abstract Properties of Linear Dependence
,
1935
.
[7]
M. D. Tobey,et al.
A Graphical Representation of Matroids
,
1973
.
[8]
Thomas A. Dowling.
A characterization of the Tm graph
,
1969
.
[9]
Stephen B. Maurer.
Matroid basis graphs. II
,
1973
.
[10]
R. Cummins.
Hamilton Circuits in Tree Graphs
,
1966
.