Reconfiguration of Graph Minors

Under the reconfiguration framework, we consider the various ways that a target graph H is a minor of a host graph G, where a subgraph of G can be transformed into H by means of edge contraction (replacement of both endpoints of an edge by a new vertex adjacent to any vertex adjacent to either endpoint). Equivalently, an H-model of G is a labeling of the vertices of G with the vertices of H, where the contraction of all edges between identically-labeled vertices results in a graph containing representations of all edges in H. We explore the properties of G and H that result in a connected reconfiguration graph, in which nodes represent H-models and two nodes are adjacent if their corresponding H-models differ by the label of a single vertex of G. Various operations on G or H are shown to preserve connectivity. In addition, we demonstrate properties of graphs G that result in connectivity for the target graphs K_2, K_3, and K_4, including a full characterization of graphs G that result in connectivity for K_2-models, as well as the relationship between connectivity of G and other H-models

Reconfiguration on sparse graphs

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs

The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs

We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACE-completeness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives precise analyses of the problem with respect to pathwidth

Complexity of Token Swapping and its Variants

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

Graph Theoretical Modelling of Electrical Distribution Grids

This thesis deals with the applications of graph theory towards the electrical distribution networks that transmit electricity from the generators that produce it and the consumers that use it. Specifically, we establish the substation and bus network as graph theoretical models for this major piece of electrical infrastructure. We also generate substation and bus networks for a wide range of existing data from both synthetic and real grids and show several properties of these graphs, such as density, degeneracy, and planarity. We also motivate future research into the definition of a graph family containing bus and substation networks and the classification of that family as having polynomial expansion

Rerouting Planar Curves and Disjoint Paths

In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths ? and ? connecting the same set of terminal pairs, we aim to determine whether ? can be transformed to ? by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in ? and ? connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general

Independent-Set Reconfiguration Thresholds of Hereditary Graph Classes

Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly k vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most k fewer vertices than the initial solution. We are interested in determining the minimum value of k for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large