Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

Distributed Design for Decentralized Control using Chordal Decomposition and ADMM

We propose a distributed design method for decentralized control by exploiting the underlying sparsity properties of the problem. Our method is based on chordal decomposition of sparse block matrices and the alternating direction method of multipliers (ADMM). We first apply a classical parameterization technique to restrict the optimal decentralized control into a convex problem that inherits the sparsity pattern of the original problem. The parameterization relies on a notion of strongly decentralized stabilization, and sufficient conditions are discussed to guarantee this notion. Then, chordal decomposition allows us to decompose the convex restriction into a problem with partially coupled constraints, and the framework of ADMM enables us to solve the decomposed problem in a distributed fashion. Consequently, the subsystems only need to share their model data with their direct neighbours, not needing a central computation. Numerical experiments demonstrate the effectiveness of the proposed method.Comment: 11 pages, 8 figures. Accepted for publication in the IEEE Transactions on Control of Network System

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

On strongly chordal graphs that are not leaf powers

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem