Characterization of Line-Consistent Signed Graphs

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem 4 ("is balanced and" was missing); also minor copyeditin

Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.Comment: 22 page

Synchronization Problems in Computer Vision

The goal of \u201csynchronization\u201d is to infer the unknown states of a network of nodes, where only the ratio (or difference) between pairs of states can be measured. Typically, states are represented by elements of a group, such as the Symmetric Group or the Special Euclidean Group. The former can represent local labels of a set of features, which refer to the multi-view matching application, whereas the latter can represent camera reference frames, in which case we are in the context of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with multiple point-set registration. A related problem is that of \u201cbearing-based network localization\u201d where each node is located at a fixed (unknown) position in 3-space and pairs of nodes can measure the direction of the line joining their locations. In this thesis we are interested in global techniques where all the measures are considered at once, as opposed to incremental approaches that grow a solution by adding pieces iteratively