Finding cactus roots in polynomial time

A cactus is a connected graph in which each edge belongs to at most one cycle. A graph H is a cactus root of a graph G if H is a cactus and G can be obtained from H by adding an edge between any two vertices in H that are of distance 2 in H. We show that it is possible to test in O(n4)O(n4) time whether an n-vertex graph G has a cactus root

A Structured Systems Approach for Optimal Actuator-Sensor Placement in Linear Time-Invariant Systems

In this paper we address the actuator/sensor allocation problem for linear time invariant (LTI) systems. Given the structure of an autonomous linear dynamical system, the goal is to design the structure of the input matrix (commonly denoted by $B$) such that the system is structurally controllable with the restriction that each input be dedicated, i.e., it can only control directly a single state variable. We provide a methodology that addresses this design question: specifically, we determine the minimum number of dedicated inputs required to ensure such structural controllability, and characterize, and characterizes all (when not unique) possible configurations of the \emph{minimal} input matrix $B$. Furthermore, we show that the proposed solution methodology incurs \emph{polynomial complexity} in the number of state variables. By duality, the solution methodology may be readily extended to the structural design of the corresponding minimal output matrix (commonly denoted by $C$) that ensures structural observability.Comment: 8 pages, submitted for publicatio

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2

Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

Finite automata and algebraic extensions of function fields

We give an automata-theoretic description of the algebraic closure of the rational function field F_q(t) over a finite field, generalizing a result of Christol. The description takes place within the Hahn-Mal'cev-Neumann field of "generalized power series" over F_q. Our approach includes a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, as well as some techniques for computing in the algebraic closure; these include an adaptation to positive characteristic of Newton's algorithm for finding local expansions of plane curves. We also conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version, includes minor edit

A Linear Time Algorithm for Computing Longest Paths in Cactus Graphs

ACM Computing Classification System (1998): G.2.2.We propose an algorithm that computes the length of a longest path in a cactus graph. Our algorithm can easily be modified to output a longest path as well or to solve the problem on cacti with edge or vertex weights. The algorithm works on rooted cacti and assigns to each vertex a two-number label, the first number being the desired parameter of the subcactus rooted at that vertex. The algorithm applies the divide-and-conquer approach and computes the label of each vertex from the labels of its children. The time complexity of our algorithm is linear in the number of vertices, thus improving the previously best quadratic time algorithm.The work performed by this author was partially funded by the Romanian National Council for Scientific Research (CNCS)-UEFISCDI under research grant PD_240/2010 (AATOMMS – contract no. 33/28.07.2010), from the PN II – RU program, and by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the financial agreement POSDRU/89/1.5/S/62557