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Pasch trades on the projective triple system of order 31
We determine all 120 nonisomorphic systems obtainable from the projective Steiner triple system of order 31 by at most three Pasch trades. Exactly three of these, each corresponding to three Pasch trades, are rigid. Thus three Pasch trades suffice, and are required, in order to convert the projective system of order 31 to a rigid system. This contrasts with the projective system of order 15 where four Pasch trades are required. We also show that four Pasch trades are required in order to convert the projective system of order 63 to a rigid system
Independence Number in Path Graphs
In the paper we present results, which allow us to compute the independence numbers of -path graphs and -path graphs of special graphs. As and are subgraphs of iterated line graphs and , respectively, we compare our results with the independence numbers of corresponding iterated line graphs
A note on the metric and edge metric dimensions of 2-connected graphs
For a given graph , the metric and edge metric dimensions of ,
and , are the cardinalities of the smallest possible
subsets of vertices in such that they uniquely identify the vertices and
the edges of , respectively, by means of distances. It is already known that
metric and edge metric dimensions are not in general comparable. Infinite
families of graphs with pendant vertices in which the edge metric dimension is
smaller than the metric dimension are already known. In this article, we
construct a 2-connected graph such that and
for every pair of integers , where . For this we use
subdivisions of complete graphs, whose metric dimension is in some cases
smaller than the edge metric dimension. Along the way, we present an upper
bound for the metric and edge metric dimensions of subdivision graphs under
some special conditions.Comment: 12 page
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