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

Enumeration of polyhedral graphs

This thesis is concerned with the design of a polyhedron enumeration algorithm. The approach taken focuses on specic classes of polyhedra and their graph theoretic properties. This is then compared more broadly to other graph enumeration algorithms that are concerned with the same or a superset which includes these properties. An original and novel algorithm is contributed to this area. The approach taken divides the problem into prescribed vertex and face degree sequences for the graphs. Using a range of existence, ordered enumeration and isomorphism techniques, it finds all unique 4-regular, 3-connected planar graphs. The algorithm is a vertex addition algorithm which means that each result output at a given stage has a new vertex added. Other results from different stages are never required for further computation and comparison, hence the process is embarrassingly parallel. Therefore, the enumeration can be distributed optimally across a cluster of computers. This work has led to a successfully implemented algorithm which takes a different approach to its treatment of the class of 4-regular, 3-connected planar graphs. As such this has led to observations and theory about other classes of graphs and graph embeddings which relate to this research

Critical percolation on random regular graphs

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random $d$-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.Comment: 10 page

Cyclic inclusion-exclusion

Following the lead of Stanley and Gessel, we consider a morphism which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph. We describe the kernel of this morphism, using a simple combinatorial operation that we call cyclic inclusion-exclusion. Our result also holds for the natural noncommutative analog and for the commutative and noncommutative restrictions to bipartite graphs. An application to the theory of Kerov character polynomials is given.Comment: comments welcom