Graph-realization problems
Abstract
A $\{$0,1$\}$-matrix M is tree graphic if there exists a tree T such that the edges of T are indexed on the rows of M and the columns of M are the incidence vectors of the edge sets of paths of T. If such a tree T exists, then T is an tree realization for M. It is vertex-tree graphic if there exists a tree T such that the vertices of T are indexed on the rows of M and the columns of M are the incidence vectors of the vertex sets of paths of T. If such a tree T exists, then T is a vertex-tree realization for M. A $\{$0,1$\}$-matrix is ditree graphic if there exists a ditree T such that the arcs of T are indexed on the rows of M and the columns of M are the incidence vectors of the arc sets of dipaths of T. If such a ditree T exists, then T is a ditree realization for M. It is vertex-ditree graphic if there exists a ditree T such that the vertices of T are indexed on the rows of M and the columns of M are the incidence vectors of the vertex sets of dipaths of T. If such a ditree T exists, then T is a vertex-ditree realization for M. A $\{$0,1$\}$-matrix M is arborescence graphic if it has a ditree realization T that is an arborescence. If such a ditree T exists, then T is an arborescence realization for M. It is vertex-arborescence graphic if it has a vertex-ditree realization T that is an arborescence. If such a ditree T exists, then T is a vertex-arborescence realization for M. The tree-realization problem is to determine whether a given $\{$0,1$\}$-matrix is tree graphic and, if so, to construct one such tree realization. The realization problems for the remaining five classes of matrices are defined analogously. In this dissertation, algorithmic and structural properties of the latter five realization problems are studied. Moreover, a polynomial-time algorithm for recognizing a class of propositional clauses that generalizes Horn sets, is presented. As a related result, a vertex analogue of 2-isomorphism for connected graphs is defined, and a vertex analogue of Whitney's theorem on 2-isomorphism is stated and proved. Also, it is shown that unlike in the edge analogue case, the set of fundamental vertex-cycles does not determine the set of vertex-cycles of a graph-tree pair, and a vertex analogue of Whitney's theorem restricted to graph-tree pairs having the same set of fundamental cycles and whose trees are paths, is stated and proved. Finally, a decomposition for 2-edge-connected graphs is presented.
Degree
Ph.D.
Advisors
Lin, Purdue University.
Subject Area
Electrical engineering|Mathematics|Computer science
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