Theta Graph Designs Anthony D. Forbes Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, UK. author Terry S. Griggs Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, UK. author Tamsin J. Forbes 22 St Albans Road, Kingston upon Thames KT2 5HQ, UK author text article 2017 eng We solve the design spectrum problem for all theta graphs with 10, 11, 12, 13, 14 and 15 edges. Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 1 16 https://jac.ut.ac.ir/article_7963_0512bf59e0fecd0dc77de2de97c2c8c6.pdf Remainder Cordial Labeling of Graphs R. Ponraj Department of Mathematics, Sri Paramakalyani College, Alwarkurichi{627 412, India author K. Annathurai Department of Mathematics, Thiruvalluvar College,, Papanasam{627 425, India author R. Kala Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli{ 627 012, India. author text article 2017 eng In this paper we introduce remainder cordial labeling of graphs. Let $G$ be a $(p,q)$ graph. Let $f:V(G)\rightarrow \{1,2,...,p\}$ be a $1-1$ map. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ or $f(v)$ is divided by $f(u)$ according as $f(u)\geq f(v)$ or $f(v)\geq f(u)$. The function$f$ is called a remainder cordial labeling of $G$ if $\left| e_{f}(0) - e_f(1) \right|\leq 1$ where $e_{f}(0)$ and $e_{f}(1)$ respectively denote the number of edges labelled with even integers and odd integers. A graph $G$ with a remainder cordial labeling is called a remainder cordial graph. We investigate the remainder cordial behavior of path, cycle, star, bistar, crown, comb, wheel, complete bipartite $K_{2,n}$ graph. Finally we propose a conjecture on complete graph $K_{n}$. Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 17 30 https://jac.ut.ac.ir/article_7965_9bf96b3c6575b2e98518384f18e0de66.pdf Asteroidal number for some product graphs S. ALAGU Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli - 627 012, Tamilnadu, India author R. KALA Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli - 627 012, Tamilnadu, India author text article 2017 eng The notion of Asteroidal triples was introduced by Lekkerkerker and Boland . D.G.Corneil and others , Ekkehard Kohler  further investigated asteroidal triples. Walter generalized the concept of asteroidal triples to asteroidal sets . Further study was carried out by Haiko Muller . In this paper we find asteroidal numbers for Direct product of cycles, Direct product of path and cycle, Strong product of paths and cycles and some more graphs. Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 31 43 https://jac.ut.ac.ir/article_7979_01beaa9142c979dd177ee0b4e047fdf0.pdf Edge-tenacity in Networks Dara Moazzami University of Tehran, College of Engineering, Department of Engineering Science author text article 2017 eng Numerous networks as, for example, road networks, electrical networks and communication networks can be modeled by a graph. Many attempts have been made to determine how well such a network is "connected" or stated differently how much effort is required to break down communication in the system between at least some nodes. Two well-known measures that indicate how "reliable" a graph is are the "Tenacity" and "Edge-tenacity" of a graph. In this paper we present results on the tenacity and edge-tenacity, $T_e(G)$, a new invariant, for several classes of graphs. Basic properties and some bounds for edge-tenacity, $T_e(G)$, are developed. Edge-tenacity values for various classes of graphs are calculated and future work andconcluding remarks are summarized Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 45 53 https://jac.ut.ac.ir/article_7980_d39c785dea8c5952d0d192f44c767675.pdf Linear optimization on the intersection of two fuzzy relational inequalities defined with Yager family of t-norms Amin Ghodousian Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran author Reza Zarghani School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, 11155-4563, Iran author text article 2017 eng In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Yager family of t-norms is considered as fuzzy composition. Yager family of t-norms is a parametric family of continuous nilpotent t-norms which is also one of the most frequently applied one. This family of t-norms is strictly increasing in its parameter and covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Yager composition. Based on some theoretical results, conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. It is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Yager fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 55 82 https://jac.ut.ac.ir/article_7981_efef0843919600ca87fd333cdebadf64.pdf Tenacity and some related results Dara Moazzami University of Tehran, College of Engineering, Department of Engineerng Science author text article 2017 eng Conceptually graph vulnerability relates to the study of graphintactness when some of its elements are removed. The motivation forstudying vulnerability measures is derived from design and analysisof networks under hostile environment. Graph tenacity has been anactive area of research since the the concept was introduced in1992. The tenacity T(G) of a graph G is defined as\begin{center} $T(G)=\displaystyle \min_{A\subset V(G)}\{\frac{\mid A\mid +\tau(G-A)}{\omega(G-A)}\}$\end{center}where $\tau(G-A)$ denotes the order (the number of vertices) of alargest component of G-A and $\omega(G-A)$ is the number ofcomponents of G-A. In this paper we discuss tenacity and its properties invulnerability calculation. Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 83 91 https://jac.ut.ac.ir/article_7986_4ca995acf8ce801abe8eb3b4123a284c.pdf An improved algorithm to reconstruct a binary tree from its inorder and postorder traversals Niloofar Aghaieabiane Department of Engineering, School of Computer Science, New Jersey Institute of Technology, Newark, New Jersey, the USA. author Henk Koppelaar Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands. author Peyman Nasehpour Golpayegan University of Technology, Department of Engineering Science, Golpayegan, Iran. author text article 2017 eng It is well-known that, given inorder traversal along with one of the preorder or postorder traversals of a binary tree, the tree can be determined uniquely. Several algorithms have been proposed to reconstruct a binary tree from its inorder and preorder traversals. There is one study to reconstruct a binary tree from its inorder and postorder traversals, and this algorithm takes running time of  $\BigO{\emph{n}^2}$. In this paper, we present $\proc{InPos}$ an improved algorithm to reconstruct a binary tree from its inorder and postorder traversals. The running time and space complexity of the algorithm are an order of $\BigTheta{\emph{n}}$ and $\BigTheta{\emph{n}}$ respectively, which we prove to be optimal.  The $\proc{InPos}$ algorithm not only reconstructs the binary tree, but also it determines different types of the nodes in a binary tree; nodes with two children, nodes with one child, and nodes with no child. At the end, the $\proc{InPos}$ returns a matrix-based structure to represent the binary tree, and enabling  access to any structural information of the reconstructed tree in linear time with any given tree traversals.   Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 93 113 https://jac.ut.ac.ir/article_7987_96afa0a5a26a75bad5082fc1b7e92603.pdf Linear optimization on Hamacher-fuzzy relational inequalities Amin Ghodousian Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran author Mohammadsadegh Nouri Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran. author text article 2017 eng In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Hamacher family of t-norms is considered as fuzzy composition. Hamacher family of t-norms is a parametric family of continuous strict t-norms, whose members are decreasing functions of the parameter. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Hamacher composition. Based on some theoretical results, a necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. It is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Hamacher fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms. Journal of Algorithms and Computation University of Tehran 2476-2776 49 v. 1 no. 2017 115 150 https://jac.ut.ac.ir/article_7988_dbb5f79575a0d1ab22f76d1af7278869.pdf