arXiv subjects

Arash Ahadi

Publications and source records attributed to Arash Ahadi.

16 recordsLinked to original sources

arXiv2025

Improved Wake-Up Time For Euclidean Freeze-Tag Problem

The Freeze-Tag Problem (FTP) involves activating a set of initially asleep robots as quickly as possible, starting from a single awake robot. Once activated, a robot can assist in waking up other robots. Each active robot moves at unit speed. The objective is to minimize the makespan, i.e., the time required to activate the last robot. A key performance measure is the wake-up ratio, defined as the maximum time needed to activate any number of robots in any primary positions. This work focuses on the geometric (Euclidean) version of FTP in $\mathbb{R}^d$ under the $\ell_p$ norm, where the initial distance between each asleep robot and the single active robot is at most 1. For $(\mathbb{R}^2, \ell_2)$, we improve the previous upper bound of 4.62 ([7], CCCG 2024) to 4.31. Note that it is known that 3.82 is a lower bound for the wake-up ratio. In $\mathbb{R}^3$, we propose a new strategy that achieves a wake-up ratio of 12 for $(\mathbb{R}^3, \ell_1)$ and 12.76 for $(\mathbb{R}^3, \ell_2)$, improving upon the previous bounds of 13 and $13\sqrt{3}$, respectively, reported in [2].

cs.CG↗

arXiv2023

Optimal Sensor Deception to Deviate from an Allowed Itinerary

In this work, we study a class of deception planning problems in which an agent aims to alter a security monitoring system's sensor readings so as to disguise its adversarial itinerary as an allowed itinerary in the environment. The adversarial itinerary set and allowed itinerary set are captured by regular languages. To deviate without being detected, we investigate whether there exists a strategy for the agent to alter the sensor readings, with a minimal cost, such that for any of those paths it takes, the system thinks the agent took a path within the allowed itinerary. Our formulation assumes an offline sensor alteration where the agent determines the sensor alteration strategy and implement it, and then carry out any path in its deviation itinerary. We prove that the problem of solving the optimal sensor alteration is NP-hard, by a reduction from the directed multi-cut problem. Further, we present an exact algorithm based on integer linear programming and demonstrate the correctness and the efficacy of the algorithm in case studies.

cs.RO↗

arXiv2021

On the maximum number of non attacking rooks on a high-dimensional simplicial chessboard

The simplicial rook graph ${\rm \mathcal{SR}}(m,n)$ is the graph whose vertices are vectors in $ \mathbb{N}^m$ such that for each vector the summation of its coordinates is $n$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin. (2015) 31:1589--1611) asked about the independence number of ${\rm \mathcal{SR}}(m,n)$ that is the maximum number of non attacking rooks which can be placed on a $(m-1)$-dimensional simplicial chessboard of side length $n+1$. In this work, we solve this problem and show that $α({\rm \mathcal{SR}}(m,n))=\big(1-o(1)\big)\frac{\binom{n+m-1}{n}}{m}$. We also prove that for the domination number of rook graphs we have $γ({\rm \mathcal{SR}}(m, n))= Θ(n^{m-2})$. Moreover we show that these graphs are Hamiltonian. The cyclic simplicial rook graph ${\rm \mathcal{CSR}}(m,n)$ is the graph whose vertices are vectors in $\mathbb{Z}^{m}_{n}$ such that for each vector the summation of its coordinates modulo $n$ is $0$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given ${\rm \mathcal{CSR}}(m,n)$ is $ \mathbf{NP}$-hard in terms of $n$ and $m$.

math.CO↗

arXiv2019

$(2/2/3)$-SAT problem and its applications in dominating set problems

The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is $\mathbf{NP}$-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is $ \mathbf{NP} $-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math., 160(15):2142--2146, 2012] to determine whether for a given independent set $T $ there is an independent dominating set $D$ that dominates $T$ such that $ T \cap D =\varnothing $? As an application of $(2/2/3)$-SAT problem we show that for every $r\geq 3$, this problem is $ \mathbf{NP} $-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $ \mathbf{NP} $-complete.

cs.DM↗

arXiv2019

Sigma Partitioning: Complexity and Random Graphs

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $σ(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $η(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ η(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.

math.CO↗

arXiv2018

On the algorithmic complexity of decomposing graphs into regular/irregular structures

A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their degrees are equal. In this work, we study the set of all problems which are related to decomposition of graphs into regular, locally regular and/or locally irregular subgraphs and we present some polynomial time algorithms, NP-completeness results, lower bounds and upper bounds for them. Among our results, one of our lower bounds makes use of mutually orthogonal Latin squares which is relatively novel.

cs.DM↗

arXiv2018

Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications

A Not-All-Equal (NAE) decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts such that each vertex in $G$ has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts $A$ and $B$ such that each vertex in the graph $G$ has exactly one neighbor in part $A$. Among our results, we show that for a given graph $G$, if $G$ does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether $G$ has a 1-in-Degree decomposition. In sharp contrast, we prove that for every $r$, $r\geq 3$, for a given $r$-regular bipartite graph $G$ determining whether $G$ has a 1-in-Degree decomposition is $ \mathbf{NP} $-complete. These complexity results have been especially useful in proving $ \mathbf{NP} $-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph $G$ determining whether there is a vector in the null-space of the 0,1-adjacency matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP} $-complete. Among other results, we introduce a new version of {Planar 1-in-3 SAT} and we prove that this version is also $ \mathbf{NP} $-complete. In consequence of this result, we show that for a given planar $(3,4)$-semiregular graph $G$ determining whether there is a vector in the null-space of the 0,1-incidence matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP} $-complete.

cs.DM↗

arXiv2017

Is there any polynomial upper bound for the universal labeling of graphs?

A {\it universal labeling} of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ and $u$ in the oriented graph are different from each other. The {\it universal labeling number} of a graph $G$ is the minimum number $k$ such that $G$ has {\it universal labeling} from $\{1,2,\ldots, k\}$ denoted it by $\overrightarrow{χ_{u}}(G) $. We have $2Δ(G)-2 \leq \overrightarrow{χ_{u}} (G)\leq 2^{Δ(G)}$, where $Δ(G)$ denotes the maximum degree of $G$. In this work, we offer a provocative question that is:" Is there any polynomial function $f$ such that for every graph $G$, $\overrightarrow{χ_{u}} (G)\leq f(Δ(G))$?". Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree $T$, $\overrightarrow{χ_{u}}(T)=\mathcal{O}(Δ^3) $. Next, we show that for a given 3-regular graph $G$, the universal labeling number of $G$ is 4 if and only if $G$ belongs to Class 1. Therefore, for a given 3-regular graph $G$, it is an $ \mathbf{NP} $-complete to determine whether the universal labeling number of $G$ is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.

math.CO↗

arXiv2017

Algorithmic complexity of proper labeling problems

A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them.

cs.DM↗

arXiv2017

Algorithmic Complexity of Weakly Semiregular Partitioning and the Representation Number

A graph $G$ is {\it weakly semiregular} if there are two numbers $a,b$, such that the degree of every vertex is $a$ or $b$. The {\it weakly semiregular number} of a graph $G$, denoted by $wr(G)$, is the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether $ wr(G) = 2 $ for a given bipartite graph $ G $ with at most three numbers in its degree set is {\bf NP}-complete. Among other results, for every tree $T$, we show that $wr(T)\leq 2\log_2 Δ(T) + \mathcal{O}(1)$, where $Δ(T)$ denotes the maximum degree of $T$. In the second part of the work, we consider the representation number. A graph $G$ has a {\it representation modulo $r$} if there exists an injective map $\ell: V (G) \rightarrow \mathbb{Z}_r$ such that vertices $v$ and $u$ are adjacent if and only if $|\ell(u) -\ell(v)|$ is relatively prime to $r$. The {\it representation number}, denoted by $rep(G)$, is the smallest $r$ such that $G$ has a representation modulo $r$. Narayan and Urick conjectured that the determination of $rep (G)$ for an arbitrary graph $G$ is a difficult problem \cite{narayan2007representations}. In this work, we confirm this conjecture and show that if $\mathbf{NP\neq P}$, then for any $ε>0$, there is no polynomial time $(1-ε)\frac{n}{2}$-approximation algorithm for the computation of representation number of regular graphs with $n$ vertices.

math.CO↗

arXiv2016

The inapproximability for the (0,1)-additive number

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ is joined to $y$). The {\it additive number} of $ G $, denoted by $η(G)$, is the minimum number $k $ such that $ G $ has a additive labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $η_{\ell}(G) $, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $η(G)= η_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $ η_{\ell}(G)- η(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $ \ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $. A graph may lack any $(0,1)$-additive labeling. We show that it is $ \mathbf{NP} $-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $ σ_{1} (G) = \min_{\ell \in Γ}\sum_{v\in V(G)}\ell(v) $ where $Γ$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $ \varepsilon >0 $, approximating the $(0,1)$-additive number within $ n^{1-\varepsilon} $ is $ \mathbf{NP} $-hard.

math.CO↗

arXiv2015

Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $ k $ colors, is called the {\it 2-hued chromatic number} of $G$ and denoted by $χ_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $ χ_{2}(G)- χ(G) \leq 2 \log _{2}(α(G)) +\mathcal{O}(1) $ and if $G$ is a graph and $δ(G)\geq 2$, then $ χ_{2}(G)- χ(G) \leq 1+\lceil \sqrt[δ-1]{4Δ^{2}} \rceil ( 1+ \log _{\frac{2Δ(G)}{2Δ(G)-δ(G)}} (α(G)) ) $ and in general case if $G$ is a graph, then $ χ_{2}(G)- χ(G) \leq 2+ \min \lbrace α^{\prime}(G),\frac{α(G)+ω(G)}{2}\rbrace $.

math.CO↗

arXiv2014

On the complexity of deciding whether the regular number is at most two

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an open problem in A. Ganesan et al. (2012) [3] about the complexity of determining the regular number of graphs. We show that computation of the regular number for connected bipartite graphs is NP-hard. Furthermore, we show that, determining whether reg(G) = 2 for a given connected 3-colorable graph G is NP-complete. Also, we prove that a new variant of the Monotone Not-All-Equal 3-Sat problem is NP-complete.

math.CO↗

arXiv2013

The Complexity of the Proper Orientation Number

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph $G = (V,E)$ is an orientation $D$ of $E(G)$ such that for every two adjacent vertices $ v $ and $ u $, $ d^{-}_{D}(v) \neq d^{-}_{D}(u)$ where $d_{D}^{-}(v)$ is the number of edges with head $v$ in $D$. The proper orientation number of $G$ is defined as $ \overrightarrowχ (G) =\displaystyle \min_{D\in Γ} \displaystyle\max_{v\in V(G)} d^{-}_{D}(v) $ where $Γ$ is the set of proper orientations of $G$. We have $ χ(G)-1 \leq \overrightarrowχ (G)\leq Δ(G) $. We show that, it is $ \mathbf{NP} $-complete to decide whether $\overrightarrowχ(G)=2$, for a given planar graph $G$. Also, we prove that there is a polynomial time algorithm for determining the proper orientation number of 3-regular graphs. In sharp contrast, we will prove that this problem is $ \mathbf{NP} $-hard for 4-regular graphs.

cs.CC↗

arXiv2011

On Rainbow Connection of Strongly Regular Graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. We prove if $G$ is a connected strongly $r$-regular graph and $r\geq 600$, then $rc(G)\leq3$. Specially, there is a constant $c$ such that $rc(G)\leq c$ for any connected strongly regular graph $G$.

math.CO↗

arXiv2010

On the Lucky labeling of Graphs

Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is, if we have $Sum(u) \neq Sum(v)$ whenever $u$ and $v$ are adjacent. The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\lbrace 1, 2, ...,k\rbrace$ is the lucky number of $G$, denoted by $η(G)$. We will prove, for every graph $G$ other than $ K_{2} $, $\frac{w}{n-w+1}\leqη(G) \leq Δ^{2} $ and we present an algorithm for lucky labeling of $ G $.

math.CO↗