Cactus Graphs Research Articles

Bounds of nullity for complex unit gain graphs

A complex unit gain graph, or T-gain graph, is a triple Φ=(G,T,φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T={z∈C:|z|=1}, and a gain function φ:E→→T with the property that φ(eij)=φ(eji)−1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n(G)−2m(G)+2c(G)−1, where n(G), m(G) and c(G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].

Reverse Zagreb Indices and Their Application in the Evaluation of Physiochemical Properties of Anticancer/Antibacterial Drugs.

Recent advancements in chemical graph theory have facilitated a deeper understanding of the relationship between chemical structures and their underlying graphs based upon classical graph theory principles. Quantitative structure-property relationship (QSPR) analysis stands out as a powerful tool for probing chemical graphs. Topological indices, graph invariants assigning numerical values to graphs, play a pivotal role in statistically correlating physical properties, chemical reactivity, and biological activity across diverse chemical structures. The cactus graph (CG) represents a noteworthy family of connected graphs distinguished by the absence of common vertices between cycles. In this study, we focus on establishing the expressions of Zagreb and reverse Zagreb indices in terms of known parameters for cactus graphs. Subsequently, we leverage these indices to conduct QSPR analysis, employing linear and multilinear regression models to investigate the relationship between the properties of chemical structures adorned with cactus-like graphs and the corresponding Zagreb and reverse Zagreb indices.

Interval colourable orientations of graphs

A graph is interval colourable if it admits a proper edge colouring in which for each vertex the set of colours of edges that are incident with the vertex is an interval of integers. An oriented graph is interval colourable if it admits a proper arc colouring in which for each vertex the set of colours of in-arcs and the set of colours of out-arcs that are incident with the vertex are both intervals of integers. In this paper we apply a classical approach and a new special trail technique to confirm the existence of interval colourable orientations of graphs from three classes: i) all k-trees with Δ(G)≤2k and k∈{2,3,4} and all k-paths with arbitrary k∈N; ii) graphs in which the length of every even closed trail is at least 2s and each such a trail has common edges with no more than d other even closed trails and 21−2se(d+1)≤1; iii) graphs that are decomposable into a bipartite interval colourable graph and a graph whose each connected component has at most one cycle that, if it exists, has odd length. Consequently, cactus graphs, graphs that are homeomorphic to Halin graphs, full subdivision graphs and others have interval colourable orientations.

The doubly metric dimensions of cactus graphs and block graphs

The doubly metric dimensions of cactus graphs and block graphs

Independence number and spectral radius of cactus graphs

Four Color Theorem implies that the independence number of every planar graph is at least n/4. A cactus graph is a connected planar graph whose block is either an edge or a cycle. In this paper, the lower bound for independence number of cactus graphs is improved to ⌈n/3⌉. Moreover, the maximum spectral radius of cactus graphs with fixed independence number is determined.

Playing Games with Cacti

Summary The Game of Cycles is a two-player impartial mathematical game, introduced by Francis Su in his book Mathematics for Human Flourishing (2020). The game is played on simple planar graphs in which players take turns marking edges using a sink-source rule. In Alvarado et al., the authors determine who is able to win on graphs with certain types of symmetry using a mirror-reverse strategy. In this paper, we analyze the game for specific types of cactus graphs using a modified version of the mirror-reverse strategy.

New Bounds for Three Outer-Independent Domination-Related Parameters in Cactus Graphs

Let G be a nontrivial connected graph. For a set D⊆V(G), we define D¯=V(G)∖D. The set D is a total outer-independent dominating set of G if |N(v)∩D|≥1 for every vertex v∈V(G) and D¯ is an independent set of G. Moreover, D is a double outer-independent dominating set of G if |N[v]∩D|≥2 for every vertex v∈V(G) and D¯ is an independent set of G. In addition, D is a 2-outer-independent dominating set of G if |N(v)∩D|≥2 for every vertex v∈D¯ and D¯ is an independent set of G. The total, double or 2-outer-independent domination number of G, denoted by γtoi(G), γ×2oi(G) or γ2oi(G), is the minimum cardinality among all total, double or 2-outer-independent dominating sets of G, respectively. In this paper, we first show that for any cactus graph G of order n(G)≥4 with k(G) cycles, γ2oi(G)≤n(G)+l(G)2+k(G), γtoi(G)≤2n(G)−l(G)+s(G)3+k(G) and γ×2oi(G)≤2n(G)+l(G)+s(G)3+k(G), where l(G) and s(G) represent the number of leaves and the number of support vertices of G, respectively. These previous bounds extend three known results given for trees. In addition, we characterize the trees T with γ×2oi(T)=γtoi(T). Moreover, we show that γ2oi(T)≥n(T)+l(T)−s(T)+12 for any tree T with n(T)≥3. Finally, we give a constructive characterization of the trees T that satisfy the equality above.

On the harmonic index in cactus graphs

The harmonic index of graph [Formula: see text] is the value [Formula: see text], where [Formula: see text] refers to the degree of [Formula: see text]. Zhong [The harmonic index for graphs, Appl. Math. Lett. 25 (2012) 561–566] proved that [Formula: see text] for any tree [Formula: see text] of order [Formula: see text]. As a results of Ali, Raza and Bhatti [Some vertex-degree-based topological indices of cacti, Ars Combin. 144 (2019) 195–206], it can be shown that [Formula: see text] for any tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, where [Formula: see text]. In this paper, we generalize this lower bound for all cactus graphs. We present a lower bound for the harmonic index of cactus graphs [Formula: see text] in terms of the order, the number of leaves and the number of cycles, and characterize all cactus graphs achieving equality for the given bound.

On the sizes of generalized cactus graphs

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a k-cactus is a connected graph in which each edge is contained in at most k cycles where k≥1. It is well known that every cactus with n vertices has at most ⌊32(n−1)⌋ edges. Inspired by it, we attempt to establish analogous upper bounds for general k-cactus graphs. In this paper, we first characterize k-cactus graphs for 2≤k≤4 based on the block decompositions. Subsequently, we give tight upper bounds on their sizes. Moreover, the corresponding extremal graphs are also characterized. However, for larger k, this extremal problem remains unsolved. Besides, we prove that every 2-connected k-cactus (k≥1) with n vertices has at most n+k−1 edges, and the bound is tight if n≥k+2.

2-tone coloring of cactus graphs

A 2-tone coloring of a graph assigns two distinct colors to each vertex with the restriction that adjacent vertices have no common colors, and vertices at distance two have at most one common color. The 2-tone chromatic number of a graph is the minimum number of colors in any 2-tone coloring. A cactus graph has every block a cycle or edge. We determine the 2-tone chromatic number of all cactus graphs with maximum degree Δ≠6. When Δ=6, we determine the 2-tone chromatic number of all cactus graphs that are triangle-free and those with circumference at most 8.

Special cases of the minimum spanning tree problem under explorable edge and vertex uncertainty

AbstractThis article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.

Sharp Lower Bound of Cacti Graph with respect to Zagreb Eccentricity Indices

The first Zagreb eccentricity index E1(℧) is the sum of square of eccentricities of the vertices, and the second Zagreb eccentricity index E2(℧) is the sum of product squares of the eccentricities of the vertices. A linked graph G is called a cactus if any two of its cycles share only one vertex. In other words, there are no two independent cycles that share an edge. Cactus graphs are also known as “block graphs” or “sensitized graphs.” They are closely related to chordal graphs and can be used to represent various types of networks, including communication networks and road networks. In this contribution, E1(℧) and E2(℧) values of cacti with k pendant vertices and k cycles, respectively, are considered. We determine the minimum E1, E2 indices for n order cacti with k pendant vertices and k cycles.

On the edge metric dimension of some classes of cacti

<abstract><p>The cactus graph has many practical applications, particularly in radio communication systems. Let $ G = (V, E) $ be a finite, undirected, and simple connected graph, then the edge metric dimension of $ G $ is the minimum cardinality of the edge metric generator for $ G $ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices $ \mathcal{G}_e = \{g_1, g_2, ..., g_k \} $ of a connected graph $ G $, for any edge $ e\in E $, we referred to the $ k $-vector (ordered $ k $-tuple), $ r(e|\mathcal{G}_e) = (d(e, g_1), d(e, g_2), ..., d(e, g_k)) $ as the edge metric representation of $ e $ with respect to $ G_e $. In this regard, $ \mathcal{G}_e $ is an edge metric generator for $ G $ if, and only if, for every pair of distinct edges $ e_1, e_2 \in E $ implies $ r (e_1 |\mathcal{G}_e) \neq r (e_2 |\mathcal{G}_e) $. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $ \mathfrak{C}(n, c, r) $ and $ \mathfrak{C}(n, m, c, r) $ in terms of the number of cycles $ (c) $ and the number of paths $ (r) $.</p></abstract>

Approximation algorithms for the graph burning on cactus and directed trees

Given a graph [Formula: see text], the problem of Graph Burning is to find a sequence of nodes from [Formula: see text], called a burning sequence, to burn the whole graph. This is a discrete-step process, and at each step, an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A burnt node spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The Graph Burning problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a [Formula: see text]-approximation algorithm for general graphs and a [Formula: see text]-approximation algorithm for trees. The Graph Burning on directed graphs is more challenging than on undirected graphs. In this paper, we propose (1) A [Formula: see text]-approximation algorithm for a cactus graph (undirected), (2) A [Formula: see text]-approximation algorithm for multi-rooted directed trees (polytree) and (3) A [Formula: see text]-approximation algorithm for the single-rooted directed tree (arborescence). We implement all three approximation algorithms, and the results are shown for randomly generated cactus graphs, multi-rooted directed trees and single-rooted directed trees.

Magnetization of the antiferromagnetic cactus graph model with an exact dimer ground state

Magnetization of the antiferromagnetic cactus graph model with an exact dimer ground state

Local Irregularity Conjecture vs. cacti

A graph in which the two end-vertices of every edge have distinct degrees is called locally irregular. An edge coloring of a graph G such that every color induces a locally irregular subgraph of G is said to be locally irregular. A graph G which admits a locally irregular edge coloring is called colorable. The Local Irregularity Conjecture claims that 3 colors are enough for locally irregular edge coloring of any colorable graph. Not so long ago the only known graphs which do not admit a locally irregular 3-edge coloring were the non-colorable graphs, all of which are cacti. This fact motivated us to study locally irregular 3-edge colorings of cacti, which first resulted with the finding of the bow-tie graph B, a colorable cactus graph which requires 4 colors for a locally irregular edge coloring. As the graph B disproves the stated form of the Local Irregularity Conjecture, here we continue the study of locally irregular 3-edge colorings of cacti in relation to the conjecture. In this paper we confirm that all other cacti are locally irregular 3-edge colorable. This result makes us to believe that B is the only counterexample to the conjecture not only in the class of cacti, but also in general.

Correction to: Powers of the vertex cover ideals (Collect. Math. 65 (2014) 169–181)

We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.

Relating the super domination and 2-domination numbers in cactus graphs

Abstract A set D ⊆ V ( G ) D\subseteq V\left(G) is a super dominating set of a graph G G if for every vertex u ∈ V ( G ) \ D u\in V\left(G)\setminus D , there exists a vertex v ∈ D v\in D such that N ( v ) \ D = { u } N\left(v)\setminus D=\left\{u\right\} . The super domination number of G G , denoted by γ s p ( G ) {\gamma }_{sp}\left(G) , is the minimum cardinality among all super dominating sets of G G . In this article, we show that if G G is a cactus graph with k ( G ) k\left(G) cycles, then γ s p ( G ) ≤ γ 2 ( G ) + k ( G ) {\gamma }_{sp}\left(G)\le {\gamma }_{2}\left(G)+k\left(G) , where γ 2 ( G ) {\gamma }_{2}\left(G) is the 2-domination number of G G . In addition, and as a consequence of the previous relationship, we show that if T T is a tree of order at least three, then γ s p ( T ) ≤ α ( T ) + s ( T ) − 1 {\gamma }_{sp}\left(T)\le \alpha \left(T)+s\left(T)-1 and characterize the trees attaining this bound, where α ( T ) \alpha \left(T) and s ( T ) s\left(T) are the independence number and the number of support vertices of T T , respectively.

Roman κ-domination revisited

Kammerling and Volkmann [J. Korean Math. Soc. 46 (2009) 1309–1318] introduced the concept of Roman κ-domination in graphs. For a fixed positive integer κ, a function f : V(G) → {0, 1, 2} is a Roman κ-dominating function on G if every vertex valued 0 under f is adjacent to at least κ vertices valued 2 under f. In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman κ-domination by not-fixing κ. We prove upper bounds for the new variant in cactus graphs and characterize cactus graph achieving equality for the given bound. We also present a probabilistic upper bound for this variant.

Binary Coding of Resonance Graphs of Catacondensed Polyhexes

A catacondensed polyhex H is a connected subgraph of a hexagonal system such that any edge of H lies in a hexagon of H, any triple of hexagons of H has an empty intersection and the inner dual of H is a cactus graph. A perfect matching M of a catacondensed polyhex H is relevant if every cycle of the inner dual of H admits a vertex that corresponds to the hexagon which contributes three edges in M. The vertex set of the graph R˜(H) consists of all relevant perfect matchings of H, two perfect matchings being adjacent whenever their symmetric difference forms the edge set of a hexagon of H. A labeling that assigns in linear time a binary string to every relevant perfect matching of a catacondensed polyhex is presented. The introduced labeling defines an isometric embedding of R˜(H) into a hypercube.