The Perfect Matching Reconfiguration Problem

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time

Bandwidth and density for block graphs

The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete.Comment: 14 pages, 9 included figures. Note: figures did not appear in original upload; resubmission corrects thi

Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks

We initiate the study of distributed protocols over directed anonymous networks that are not necessarily strongly connected. In such networks, nodes are aware only of their incoming and outgoing edges, have no unique identity, and have no knowledge of the network topology or even bounds on its parameters, like the number of nodes or the network diameter. Anonymous networks are of interest in various settings such as wireless ad-hoc networks and peer to peer networks. Our goal is to create distributed protocols that reduce the uncertainty by distributing the knowledge of the network topology to all the nodes. We consider two basic protocols: broadcasting and unique label assignment. These two protocols enable a complete mapping of the network and can serve as key building blocks in more advanced protocols. We develop distributed asynchronous protocols as well as derive lower bounds on their communication complexity, total bandwidth complexity, and node label complexity. The resulting lower bounds are sometimes surprisingly high, exhibiting the complexity of topology extraction in directed anonymous networks