Edge-disjoint Cycles Research Articles

An Efficient Algorithm to Compute Dot Product Dimension of Some Outerplanar Graphs

A graph [Formula: see text] is called a [Formula: see text]-dot product graph if there is a function [Formula: see text] such that for any two distinct vertices [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]. The minimum value [Formula: see text] such that [Formula: see text] is a [Formula: see text]-dot product graph, is called the dot product dimension [Formula: see text] of [Formula: see text]. In this paper, we give an efficient algorithm for computing the dot product dimension of outerplanar graphs of at most two edge-disjoint cycles. If the graph has two cycles, we only consider those outerplanar graphs if both cycles have exactly one vertex in common and the length of one of the cycles is greater than or equal to six.

Rooted Prism-Minors and Disjoint Cycles Containing a Specified Edge

Dirac and Lovász independently characterized the $3$-connected graphs with no pair of vertex-disjoint cycles. Equivalently, they characterized all $3$-connected graphs with no prism-minors. In this paper, we completely characterize the $3$-connected graphs with no edge that is contained in the union of a pair of vertex-disjoint cycles. As applications, we answer the analogous questions for edge-disjoint cycles and for $4$-connected graphs and we completely characterize the $3$-connected graphs with no prism-minor using a specified edge.

A One Pass Streaming Algorithm for Finding Euler Tours

Given an undirected graph G on n nodes and m edges in the form of a data stream we study the problem of finding an Euler tour in G. Our main result is the first one-pass streaming algorithm computing an Euler tour of G in the form of an edge successor function with only mathcal O(nlog (n)) RAM, which is optimal for this setting (e.g. Sun and Woodruff (2015)). Since the output size can be much larger, we use a write-only tape to gradually output the solution. The previously best-known result for finding Euler tours in data streams is implicitly given by the W-stream algorithm of Demetrescu et al. (2010) using mathcal O(m/n) passes under the same RAM limitation. Our approach is to partition the edges into edge-disjoint cycles and to merge the cycles until a single Euler tour is achieved. In the streaming environment such a merging is far from being obvious as the limited RAM allows the processing of only a constant number of cycles at once. This enforces merging of cycles that partially are no longer present in RAM. We solve this problem with a new edge swapping technique, for which storing two certain edges per node is sufficient to merge tours without having all tour edges in RAM. The mathematical key is to model tours and their merging in an algebraic way, where certain equivalence classes represent subtours. This quite general approach might be of interest also in other routing problems.

Domination, Independence and Fibonacci Numbers in Graphs Containing Disjoint Cycles

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

Embedding mutually edge-disjoint cycles into locally twisted cubes

The n-dimensional locally twisted cube LTQn, an important variation of the hypercube, can be denoted by LTQn=LTQn−10⊕LTQn−11=(LTQn−200⊕LTQn−201)⊕(LTQn−210⊕LTQn−211). In this paper, we prove that for any node x in LTQn, there are ⌊n2⌋ edge-disjoint cycles containing x such that each cycle has intersection with each LTQn−2αβ, where αβ∈{00,01,10,11} and n≥4. Since LTQn is n-regular and each node has degree 2 in each cycle, our result is optimal.

Spectral Radius, Edge-Disjoint Cycles and Cycles of the Same Length

In this paper, we provide spectral conditions for the existence of two edge-disjoint cycles and two cycles of the same length in a graph, which can be viewed as the spectral analogues of Erdős and Posa's condition and Erdős' classic problem about the maximum number of edges of a graph without two edge-disjoint cycles and two cycles of the same length, respectively. Furthermore, we give a spectral condition to guarantee the existence of $k$ edge-disjoint triangles in a graph.

Frequency-driven market mechanisms for optimal dispatch in power networks

This paper studies real-time bidding mechanisms for economic dispatch and frequency regulation in electrical power networks described by topologies with edge-disjoint cycles. We consider a market administered by an independent system operator (ISO) where a group of strategic generators participate in a Bertrand game of competition. Generators bid prices at which they are willing to produce electricity. Each generator aims to maximize their profit, while the ISO seeks to minimize the total generation cost while respecting line flow limits and regulate the frequency of the system. We consider a continuous-time bidding process coupled with the swing dynamics of the network through the use of frequency as a feedback signal for the negotiation process. We analyze the stability of the resulting interconnected system, establishing frequency regulation and the convergence to a Nash equilibrium and optimal generation levels. Simulations illustrate our theoretical findings.

Mixed metric dimension of graphs with edge disjoint cycles

In a connected graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles, where for each cycle Ci of G we define a unicyclic subgraph Gi of G in which Ci is the only cycle. Applying the result for unicyclic graph to the subgraph Gi of every cycle Ci then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number.

Extremal mixed metric dimension with respect to the cyclomatic number

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that every graph G with cyclomatic number c(G) satisfies mdim(G)≤L1(G)+2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree ≥3. In this paper we determine that for every Θ-graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Θ-graph. Thus, for balanced Θ-graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G)=L1(G)+2c(G) holds.

Long cycles, heavy cycles and cycle decompositions in digraphs

Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean (1986) [7], and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed into O(n) cycles.In this paper, we show that every directed Eulerian graph can be decomposed into O(nlog⁡Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω(log⁡log⁡n/log⁡n), thus resolving a question by Bollobás and Scott.

Metric Dimension on Sparse Graphs and its Applications to Zero Forcing Sets

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. Our main result consists in proving that the increase of the metric dimension of an edge addition can be amortized in the sense that if the graph consists of a spanning tree $T$ plus $c$ edges, then the metric dimension of $G$ is at most the metric dimension of $T$ plus $6c$. We then use this result to prove a weakening of a conjecture of Eroh et al. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a subset $S$ of black vertices (whereas the other vertices are colored white) of $G$ such that all the vertices will turned black after applying finitely many times the following rule: a white vertex is turned black if it is the only white neighbor of a black vertex. Eroh et al. conjectured that, for any graph $G$, $dim(G)\leq Z(G) + c(G)$, where $c(G)$ is the number of edges that have to be removed from $G$ to get a forest. They proved the conjecture is true for trees and unicyclic graphs. We prove a weaker version of the conjecture: $dim(G)\leq Z(G)+6c(G)$ holds for any graph. We also prove that the conjecture is true for graphs with edge disjoint cycles, widely generalizing the unicyclic result of Eroh et al.

Bounds on metric dimensions of graphs with edge disjoint cycles

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V(G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ by at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles (i.e. cactus graphs) showing that the two invariants can differ by at most c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.

On edge-path eigenvalues of graphs

ABSTRACT Let G be a graph with vertex set and be an matrix whose -entry is the maximum number of internally edge-disjoint paths between and , if , and zero otherwise. Also, define , where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing , whose is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix for an arbitrary bicyclic graph G.

Cycle decompositions and constructive characterizations

Decomposing an Eulerian graph into a minimum respectively maximum number of edge disjoint cycles is an NP-complete problem. We prove that an Eulerian graph decomposes into a unique number of cycles if and only if it does not contain two edge disjoint cycles sharing three or more vertices. To this end, we discuss the interplay of three binary graph operators leading to novel constructive characterizations of two subclasses of Eulerian graphs. This enables us to present a polynomial-time algorithm which decides whether the number of cycles in a cycle decomposition of a given Eulerian graph is unique.

Maximum cycle packing using SPR-trees

Let G = ( V , E ) be an undirected multigraph without loops. The maximum cycle packing problem is to find a collection Z * = { C 1 , ..., C s } of edge-disjoint cycles C i subset G of maximum cardinality v ( G ) . In general, this problem is NP-hard. An approximation algorithm for computing v ( G ) for 2-connected graphs is presented, which is based on splits of G . It essentially uses the representation of the 3-connected components of G by its SPR-tree. It is proved that for generalized series-parallel multigraphs the algorithm is optimal, i.e. it determines a maximum cycle packing Z * in linear time.

Using deficit functions for aircraft fleet routing

We consider the problem of minimizing the number of airplanes needed to fly a fixed daily repeating schedule of flights. We use deficit functions (DF) to decompose an aviation schedule of aircraft flights into aircraft chains (routes) called a chain decomposition. Each chain visits periodically a set of airports and is served by several cockpit crews circulating along the airports of this set. The initial step in our approach is to find the minimal number of aircraft needed to carry out the flight schedule. This is achieved by using the fleet size theorem based on a DF representation of an aircraft flight schedule. A DF is a step function associated with an aircraft terminal which changes by + 1 and −1 at flight departure and arrival times, respectively. DF theory was developed in the 1960–70s by Linis and Maksim (1967) and Gertsbakh and Gurevich (1977). Although the initial application of DFs was to the Russian AEROFLOT fleet it has subsequently attracted more attention on bus scheduling than aircraft scheduling. Here we discuss the revival of this method and its crucial use to construct the so-called chain decomposition of the schedule for a single period. We provide a justification for maximizing the number of balanced chains (flight sequences with the same start and end terminals). To do this we propose The Maximal Balanced Chain Problem. These are then converted into a set of infinite periodic flight sequences, each of which can be carried out by a single aircraft. The conversion is carried out by mapping the single period set of chains into an Euler graph. To construct the set of mutiperiod chains that are “balanced” (return to the same terminal at the start) we find all edge disjoint cycle covers of the Euler graph using a modified version of Hierholzer's algorithm. These cycles are converted back into a balanced multiperiod chain solution and modified to conform to any maintenance constraints. To insure maintenance check constraints are satisfied for multiperiod chains, it may be necessary to add deadhead flights. Minimizing the cost of deadhead trips and overnight stays provide the basis for selecting an optimal routing solution.

Optimal information ratio of secret sharing schemes on Dutch windmill graphs

One of the basic problems in secret sharing is to determine the exact values of the information ratio of the access structures. This task is important from the practical point of view, since the security of any system degrades as the amount of secret information increases. A Dutch windmill graph consists of the edge-disjoint cycles such that all of them meet in one vertex. In this paper, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, we determine the exact ratio of some related graph families.

Long Cycles have the Edge-Erdős-Pósa Property

We prove that the set of long cycles has the edge-Erdős-Posa property: for every fixed integer l ≥ 3 and every k ∈ ℕ, every graph G either contains k edge-disjoint cycles of length at least l (long cycles) or an edge set X of size O(k2 logk+kl) such that G—X does not contain any long cycle. This answers a question of Birmele, Bondy, and Reed (Combinatorica 27 (2007), 135-145).

Finding local genome rearrangements

BackgroundThe double cut and join (DCJ) model of genome rearrangement is well studied due to its mathematical simplicity and power to account for the many events that transform gene order. These studies have mostly been devoted to the understanding of minimum length scenarios transforming one genome into another. In this paper we search instead for rearrangement scenarios that minimize the number of rearrangements whose breakpoints are unlikely due to some biological criteria. One such criterion has recently become accessible due to the advent of the Hi-C experiment, facilitating the study of 3D spacial distance between breakpoint regions.ResultsWe establish a link between the minimum number of unlikely rearrangements required by a scenario and the problem of finding a maximum edge-disjoint cycle packing on a certain transformed version of the adjacency graph. This link leads to a 3/2-approximation as well as an exact integer linear programming formulation for our problem, which we prove to be NP-complete. We also present experimental results on fruit flies, showing that Hi-C data is informative when used as a criterion for rearrangements.ConclusionsA new variant of the weighted DCJ distance problem is addressed that ignores scenario length in its objective function. A solution to this problem provides a lower bound on the number of unlikely moves necessary when transforming one gene order into another. This lower bound aids in the study of rearrangement scenarios with respect to chromatin structure, and could eventually be used in the design of a fixed parameter algorithm with a more general objective function.

Can a breakpoint graph be decomposed into none other than 2-cycles?

Breakpoint graph has been widely used as a key data structure in algorithm design for genome rearrangements. The problem of breakpoint graph cycle decomposition, which asks for a largest collection of edge-disjoint cycles, is crucial in computing rearrangement distances between genomes. This problem is NP-hard, and can be approximated to 1.4193+ϵ. It is still open for deciding whether a breakpoint graph can admit a cycle decomposition with none other than 2-cycles. In this paper, we present a linear time algorithm to detect whether a breakpoint graph can be decomposed into none other than 2-cycles.