Cactus Graphs Research Articles

Connected power domination in graphs

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.

Fair domination number in cactus graphs

Fair domination number in cactus graphs

Bounds on the weighted vertex PI index of cacti graphs

The weighted vertex PI index of a graph G is defined by PIw(G)=?e=uv?E(G) (dG(u)+dG(v))(nu(e|G) + nv(e|G)) where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes.By building on the work of Gioan and Paul (2012) and Chauve et al.(2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph.We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti); in other cases, although the enumerations were known, an abundant potential is unlocked by the grammars we provide (in terms of asymptotic analysis, random generation, and parameter analyses, etc.).We believe this methodology here shows its potential; the natural next step to develop its reach would be to study split-decomposition trees which contain certain prime nodes. This will be the object of future work.

Relating the total domination number and the annihilation number of cactus graphs and block graphs

The total domination number γ t ( G ) of a graph G is the order of a smallest set D ⊆ V ( G ) such that each vertex of G is adjacent to some vertex in D . The annihilation number a ( G ) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most | E ( G )| . It is conjectured that γ t ( G ) ≤ a ( G ) + 1 holds for every nontrivial connected graph G . The conjecture was proved for graphs with minimum degree at least 3 , and remains unresolved for graphs with minimum degree 1 or 2 . In this paper we establish the conjecture for cactus graphs and block graphs.

On the relation between the positive inertia index and negative inertia index of weighted graphs

Let Gw be an edge-weighted graph and let A(Gw) be its adjacency matrix. The positive inertia index (respectively, the negative inertia index) of Gw, denoted by pGw+ (resp. pGw−), is defined to be the number of positive eigenvalues (resp. negative eigenvalues) of A(Gw). Clearly, pGw++pGw− is the rank of Gw, whereas pGw+−pGw− is defined as the signature of Gw. In this paper, some upper and lower bounds on pGw+−pGw− are determined for the weighted chordal graphs (resp. bipartite graphs, complete graphs, cactus graphs, join graphs, bisplit graphs, power graphs).

The Hosoya polynomial of double weighted graphs

The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.

Technical Note—The Competitive Facility Location Problem in a Duopoly: Advances Beyond Trees

We consider a competitive facility location problem on a network where consumers located on vertices wish to connect to the nearest facility. Knowing this, each competitor locates a facility on a vertex, trying to maximize market share. We focus on the two-player case and study conditions that guarantee the existence of a pure-strategy Nash equilibrium for progressively more complicated classes of networks. For general graphs, we show that attention can be restricted to a subset of vertices referred to as the central block. By constructing trees of maximal bi-connected components, we obtain sufficient conditions for equilibrium existence. Moreover, when the central block is a vertex or a cycle (for example, in cactus graphs), this provides a complete and efficient characterization of equilibria. In that case, we show that both competitors locate their facilities in a solution to the 1-median problem, generalizing a well-known insight arising from Hotelling’s model. We further show that an equilibrium must solve the 1-median problem in other classes of graphs, including grids, which essentially capture the topology of urban networks. In addition, when both players select a 1-median, the solution must be at equilibrium for strongly-chordal graphs, generalizing a previously known result for trees. The electronic companion is available at https://doi.org/10.1287/opre.2017.1694 .

Cactus graphs with minimum edge revised Szeged index

The edge revised Szeged index Sze∗(G) is defined as Sze∗(G)=∑e=uv∈E(mu(e)+m0(e)∕2)(mv(e)+m0(e)∕2), where mu(e) and mv(e) are, respectively, the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, and m0(e) is the number of edges equidistant to u and v. A cactus graph is a connected graph in which every block is an edge or a cycle. In this paper, we give a lower bound of the edge revised Szeged index among all m-edges cactus graphs with k cycles, and also characterize those graphs that achieve the lower bound. We also obtain the second minimum edge revised Szeged index for connected cactus graphs of size m with k cycles.

Computation of Gutman index of some cactus chains

Let G be a finite connected graph of order n. The Gutman index Gut(G) of G is defined as $\sum_{\{x,y\}\subseteq V(G)}deg(x)deg(y)d(x, y)$, where deg(x) is the degree of vertex x in G and d(x, y) is the distance between vertices x and y in G. A cactus graph is a connected graph in which no edge lies in more than one cycle. In this paper we compute the exact value of Gutman index of some cactus chains.

Complexity of minimum irreducible infeasible subsystem covers for flow networks

For an infeasible network flow system with supplies and demands, we consider the problem of finding a minimum irreducible infeasible subsystem cover, i.e., a smallest set of constraints that must be dropped to obtain a feasible system. The special cases of covers which only contain flow balance constraints (node cover) or only flow bounds (arc cover) are investigated as well. We show strong NP-hardness of all three variants. Furthermore, we show that finding minimum arc covers for assignment problems is still hard and as hard to approximate as the set covering problem. However, the minimum arc cover problem is polynomially solvable for networks on cactus graphs. This leads to the development of two different fixed parameter algorithms with respect to the number of elementary cycles connected at arcs and the treewidth, respectively. The latter can be adapted for node covers and the general case.

Superbubbles, Ultrabubbles, and Cacti.

A superbubble is a type of directed acyclic subgraph with single distinct source and sink vertices. In genome assembly and genetics, the possible paths through a superbubble can be considered to represent the set of possible sequences at a location in a genome. Bidirected and biedged graphs are a generalization of digraphs that are increasingly being used to more fully represent genome assembly and variation problems. In this study, we define snarls and ultrabubbles, generalizations of superbubbles for bidirected and biedged graphs, and give an efficient algorithm for the detection of these more general structures. Key to this algorithm is the cactus graph, which, we show, encodes the nested decomposition of a graph into snarls and ultrabubbles within its structure. We propose and demonstrate empirically that this decomposition on bidirected and biedged graphs solves a fundamental problem by defining genetic sites for any collection of genomic variations, including complex structural variations, without need for any single reference genome coordinate system. Further, the nesting of the decomposition gives a natural way to describe and model variations contained within large variations, a case not currently dealt with by existing formats [e.g., variant cell format (VCF)].

Zeroth-order general Randić index of cactus graphs

A connected graph G is said to be cactus if no two cycles of G have any common edge. The present note is devoted to developing some extremal results for the zeroth-order general Randić index of cactus graphs and finding some sharp bounds on this index.

Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given a graph G, positive integers k and ℓ and two vertex covers S and T of G of size at most k, we determine whether S can be transformed into T by a sequence of at most ℓ vertex additions or removals such that every operation results in a vertex cover of size at most k. Motivated by results establishing the W [ 1 ] -hardness of VCR when parameterized by ℓ, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W [ 1 ] -hard on bipartite graphs, is NP -hard, but fixed-parameter tractable on (regular) graphs of bounded degree and more generally on nowhere dense graphs and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.

The Cordial Labeling for the Four-Leaved Rose Graph

A cactus graph with four blocks which are all cycles, not necessarily be of the same size, is called four-leaved rose graph and denoted by Ln, m, k, s, where n, m, k and s represent she sizes of the four cycles. A cordial graph is a graph whose vertices and edges have 0-1 labeling in such a way that the number of vertices (edges) labelled with zeros and the number of vertices (edges) labelled with ones differ absolutely by at most one .In this paper, we study this graph in detail and show that any four-leaved rose graph is cordial for all n, m, k and s except possibly at n, m are odd with (k + s) = 0(mod4) or n, m are even with (k + s) = 2(mod4). Our technique depends on the methods that partition off the set of positive integers and then use suitable labeling in each division of the partition to achieve our results. AMS classification 05C76, 05C78

On independent [1,2]-sets in trees

An independent $[1,k]$-set $S$ in a graph $G$ is a dominating set which is independent and such that every vertex not in $S$ has at most $k$ neighbors in it. The existence of such sets is not guaranteed in every graph and trees having an independent $[1,k]$-set have been characterized. In this paper we solve some problems previously posed by other authors about independent $[1,2]$-sets. We provide a necessary condition for a graph to have an independent $[1,2]$-set, in terms of spanning trees and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belong to some independent $[1,2]$-set. Finally we describe a linear algorithm to decide whether a tree has an independent $[1,2]$-set. Such algorithm can be easily modified to obtain the cardinality of the smallest independent $[1,2]$-set of a tree.

Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free group Fn and inspect their BNSR-invariants. We prove that for n≥2, all the “positive” and “negative” character classes of PΣAutn lie in Σn−2(PΣAutn)∖Σn−1(PΣAutn). We use this to prove that for n≥2, Σn−2(ΣAutn) equals the full character sphere S0 of ΣAutn but Σn−1(ΣAutn) is empty, so in particular the commutator subgroup ΣAutn' is of type Fn−2 but not Fn−1. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

A Self-Stabilizing Algorithm for a Maximal 2-Packing in a Cactus Graph Under Any Scheduler

In this paper, we present a self-stabilizing algorithm that computes a maximal 2-packing set in a cactus under the adversarial scheduler. The cactus is a network topology such that any edge belongs to at most one cycle. The cactus has important applications in telecommunication networks, location problems, and biotechnology, among others. We assume that the value of each vertex identifier can take any value of length [Formula: see text] bits. The execution time of this algorithm is [Formula: see text] rounds or [Formula: see text] time steps. Our algorithm matches the state of the art results for this problem, following an entirely different approach. Our approach allows the computation of the maximum 2-packing when the cactus is a ring.

The connected p-center problem on cactus graphs

In this paper we study a variant of the p-center problem on cactus graphs in which the p-center are asked to be connected, and this problem is called the connected p-center problem. For the connected p-center problem on cactus graphs, we propose a dynamic programming algorithm and show that the time complexity is O(n2p2), where n is the number of the vertices of the cactus graphs.

The shortest connection game

We introduce Shortest Connection Game, a two-player game played on a directed graph with edge costs. Given two designated vertices in which the players start, the players take turns in choosing edges emanating from the vertex they are currently located at. This way, each of the players forms a path that origins from its respective starting vertex. The game ends as soon as the two paths meet, i.e., a connection between the players is established. Each player has to carry the cost of its chosen edges and thus aims at minimizing its own total cost.In this work we analyse the computational complexity of Shortest Connection Game. On the negative side, Shortest Connection Game turns out to be computationally hard even on restricted graph classes such as bipartite, acyclic and cactus graphs. On the positive side, we can give a polynomial time algorithm for cactus graphs when the game is restricted to simple paths.