The Connected P -Median Problem on Cactus Graphs

This study deals with the facility location problem of locating a set V p of p facilities on a graph such that the subgraph induced by V p is connected. We consider the connected p -median problem on a cactus graph G whose vertices and edges have nonnegative weights. The aim of a connected p -median problem is to minimize the sum of weighted distances from every vertex of a graph to the nearest vertex in V p . We provide an O n 2 p 2 time algorithm for the connected p -median problem, where n is the number of vertices.

Remarks on the Local Irregularity Conjecture

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by χirr′(G), is the smallest number of colors used by a locally irregular edge coloring of G. The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χirr′(B)=4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.

Edge Mostar Indices of Cacti Graph With Fixed Cycles

Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph ℋ, the edge Mostar invariant is described as Moe(ℋ)=∑gx∈E(ℋ)|mℋ(g)−mℋ(x)|, where mℋ(g)(or mℋ(x)) is the number of edges of ℋ lying closer to vertex g (or x) than to vertex x (or g). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and n vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in ℭ(n,s), where s is the number of cycles.

Computing total edge irregularity strength of some n-uniform cactus chain graphs and related chain graphs

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Given graph </span><em>G</em><span>(</span><span><em>V</em>,<em>E</em></span><span>)</span><span>. We use the notion of total </span><em>k</em><span>-labeling which is edge irregular. The notion </span>of total edge irregularity strength (tes) of graph <em>G</em> means the minimum integer <em>k</em> used in the edge irregular total k-labeling of <em>G</em>. A cactus graph <em>G</em> is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle <em>C_n</em> with same size <em>n</em> is named an <em>n</em>-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then <em>G</em> is called <em>n</em>-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs <em>C</em>(<em>C_n^r</em>) of length <em>r</em> for some <em>n</em> ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs <em>T_r</em>(4,<em>n</em>) and <em>T_r</em>(5,<em>n</em>) of length <em>r</em>. Our results are as follows: tes(<em>C</em>(<em>C_n^r</em>)) = ⌈(<em>nr</em> + 2)/3⌉ ; tes(<em>T_r</em>(4,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉ ; tes(<em>T_r</em>(5,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉.</p></div></div></div>

On the Aα-Spectral Radii of Cactus Graphs

Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

A bottom-up algorithm for solving ♯2SAT

Counting models for a two conjunctive formula (2-CF) $F$, a problem known as $\sharp $2Sat, is a classic $\sharp $P complete problem. Given a 2-CF $F$ as input, its constraint graph $G$ is built. If $G$ is acyclic, then $\sharp $2Sat($F$) can be computed efficiently. In this paper, we address the case when $G$ has cycles. When $G$ is cyclic, we propose a decomposition on the constraint graph $G$ that allows the computation of $\sharp $2Sat($F$) in incremental way. Let $T$ be a cactus graph of $G$ containing a maximal number of independent cycles, and let $\overline{T}=(E(G)-E(T))$ be a subset of frond edges from $G$. The clauses in $\overline{T}$ are ordered in connected components $\{K_1, \ldots , K_r\}$. Each $(G \cup K_i), i=1,\ldots ,r$ is a knot (a set of intersected cycles) of the graph. The arrangement of the clauses of $\overline{T}$ allows the decomposition of $G$ in knots and provides a way of computing $\sharp $2Sat(F) in an incremental way. Our procedure has a bottom-up orientation for the computation of $\sharp $2Sat($F$). It begins with $F_0 = T$. In each iteration of the procedure, a new clause $C_i \in \overline{T}$ is considered in order to form $F_i = (F_{i-1} \wedge C_i)$ and then to compute $\sharp $2Sat$(F_i)$ based on the computation of $\sharp $2Sat$(F_{i-1})$.

On Rainbow Connection Number of Some Graphs

The Rainbow connection number for the following graphs, two copies of Fan graph by a path , Arrow graph and Θ , Jellyfish graph and Cycle Cactus graph have been described in this paper

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.