4 research outputs found
The structure of graphs with a vital linkage of order 2
A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs with a vital linkage of order 2: they are certain minors of a family of highly structured graphs
The structure of graphs with a vital linkage of order 2 *
Abstract A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs with a vital linkage of order 2: they are certain minors of a family of highly structured graphs
Rigid linkages and partial zero forcing
Connections between vital linkages and zero forcing are established.
Specifically, the notion of a rigid linkage is introduced as a special kind of
unique linkage and it is shown that spanning forcing paths of a zero forcing
process form a spanning rigid linkage and thus a vital linkage. A related
generalization of zero forcing that produces a rigid linkage via a coloring
process is developed. One of the motivations for introducing zero forcing is to
provide an upper bound on the maximum multiplicity of an eigenvalue among the
real symmetric matrices described by a graph. Rigid linkages and a related
notion of rigid shortest linkages are utilized to obtain bounds on the
multiplicities of eigenvalues of this family of matrices.Comment: 23 page