We extend graph rotation systems into a solid mathematical model for the development of an interactive-graphics cyclic-weaving system. It involves a systematic exploration and characterization of dynamic surgery operations on graph rotation systems, such as edge-insertion, edge-deletion, and edge-twisting. This talk explains the underlying mathematics and some high-level aspects of the programm...

The tau constant is an important invariant of a metrized graph. It has connections to other graph invariants such as Kirchhoff index, and it has applications to arithmetic properties of algebraic curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We prove identities which we call “contraction”, “deletion”, and “contraction-deletion...

Given a graph G and integers k1, k2, and k3, the unit interval editing problem asks whether G can be transformed into a unit interval graph by at most k1 vertex deletions, k2 edge deletions, and k3 edge additions. We give an algorithm solving this problem in time 2 logk) · (n+m), where k := k1 +k2 +k3, and n,m denote respectively the numbers of vertices and edges of G. Therefore, it is fixed-pa...

In this paper we study the problem of maintaining a single source shortest path tree or a breadth-rst search tree for a directed graph, in either an incremental or decremental setting. We maintain a single source shortest path tree of a directed graph G with unit edge weights during a sequence of edge deletions in total time O(mn), thus obtaining a O(n) amortized time for each deletion if the s...

We give a new characterization of chordal graphs via edge deletion from a complete graph, where edges eligible for deletion are each properly contained in a single maximal clique. Interestingly, in this context such edges always arise in cycles, and we use this fact to give a modified version of Kruskal’s second algorithm for finding a minimum spanning tree in a finite metric space.

We consider the Schur complement operation for symmetric matrices over GF(2), which we identify with graphs through the adjacency matrix representation. It is known that Schur complementation for such a matrix (i.e., for a graph) can be decomposed into a sequence of two types of elementary Schur complement operations: (1) local complementation on a looped vertex followed by deletion of that ver...

In the Cluster Editing problem, a graph is to be changed to a disjoint union of cliques by at most k operations of edge insertion or edge deletion. Improving on the best previously known quadratic-size polynomial-time kernelization, we describe how a crown-type structural reduction rule can be used to obtain a 6k kernelization bound.

Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually, the origin of this concept came back to Richard Brualdi's problems that are proposed in cite{braldi}: Let $G_n$ and $G'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2...

For fixed integers r, ` ≥ 0, a graph G is called an (r, `)-graph if the vertex set V (G) can be partitioned into r independent sets and ` cliques. This brings us to the following natural parameterized questions: Vertex (r, `)-Partization and Edge (r, `)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set...

In a (parameterized) graph edge modification problem, we are given G, an integer k and (usually well-structured) class $$\mathcal {G}$$ of graphs, asked whether it is possible to transform G into $$G' \in \mathcal by adding and/or removing at most edges. Parameterized problems received considerable attention in the last decades. this paper, focus on finding small kernels for problems. One studi...