Metric Dimension of Amalgamation of Graphs

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let $\{G_1, G_2, \ldots, G_n\}$ be a finite collection of graphs and each $G_i$ has a fixed vertex $v_{0_i}$ or a fixed edge $e_{0_i}$ called a terminal vertex or edge, respectively. The \emph{vertex-amalgamation} of $G_1, G_2, \ldots, G_n$, denoted by $Vertex-Amal\{G_i;v_{0_i}\}$, is formed by taking all the $G_i$'s and identifying their terminal vertices. Similarly, the \emph{edge-amalgamation} of $G_1, G_2, \ldots, G_n$, denoted by $Edge-Amal\{G_i;e_{0_i}\}$, is formed by taking all the $G_i$'s and identifying their terminal edges. Here we study the metric dimensions of vertex-amalgamation and edge-amalgamation for finite collection of arbitrary graphs. We give lower and upper bounds for the dimensions, show that the bounds are tight, and construct infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised version 21 December 201

Unlocking the Barriers of Household Waste Recycling in Seremban, Malaysia

This preliminary study investigates the barriers inhibiting effective household waste recycling in Seremban, Malaysia. Data was gathered from 101 households using a structured Google Form questionnaire over two months. Seven barriers were identified, with limited access to recycling facilities, lack of awareness, and inconvenient collection schedules among the most prominent. Quantitative statistical methods were employed to analyze the data, revealing mean scores and skewness for each barrier. The results of this initial investigation will serve as a foundation for a more comprehensive study employing the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method to analyze complex interrelationships between barriers. Keywords: waste recycling; household; Seremban; Dematel eISSN: 2398-4287 © 2023. The Authors. Published for AMER and cE-Bs by e-International Publishing House, Ltd., UK. This is an open-access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer–review under the responsibility of AMER (Association of Malaysian Environment-Behaviour Researchers), and cE-Bs (Centre for Environment-Behaviour Studies), College of Built Environment, Universiti Teknologi MARA, Malaysia. DOI: https://doi.org/10.21834/e-bpj.v8iSI15.506

Restricted Size Ramsey Number for Path of Order Three Versus Graph of Order Five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller\u27s works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five

Optimization Model for an Airline Crew Rostering Problem: Case of Garuda Indonesia

This paper discusses the cockpit crew rostering problem at Garuda Indonesia, taking into account a number of internal cockpit crew labor regulations. These internal labor regulations are in general more restrictive at Garuda Indonesia than at other airlines, so that modeling the cockpit crew rostering problem for Garuda Indonesia is challenging. We have derived mathematical expressions for the cockpit crew labor regulations and some technical matters. We model a non-linear integer programming for the rostering problem, using the average relative deviation of total flight time to the ideal flight time as the objective function. The optimization model have been tested for all classes of cockpit crews of Garuda Indonesia, using a simulated annealing method for solving the problem. We obtained satisfactory rosters for all crew members in a short amount of computing time. This shows that the optimization problem is well-defined

On the Restricted Size Ramsey Number Involving a Path P3

For any pair of graphs G and H, both the size Ramsey number ̂r(G,H) and the restricted size Ramsey number r*(G,H) are bounded above by the size of the complete graph with order equals to the Ramsey number r(G,H), and bounded below by e(G) + e(H) − 1. Moreover, trivially, ̂r(G,H) ≤ r*(G,H). When introducing the size Ramsey number for graph, Erdős et al. (1978) asked two questions; (1) Do there exist graphs G and H such that ˆr(G,H) attains the upper bound? and (2) Do there exist graphs G and H such that ̂r(G,H) is significantly less than the upper bound

Restricted Size Ramsey Number Involving Matching and Graph of Order Five

Harary and Miller (1983) started the research on the (restricted) size Ramsey number for a pair of small graphs. They obtained the values for some pairs of small graphs with order not more than four. In the same year, Faudree and Sheehan continued the research and extended the result to all pairs of small graphs with order not more than four. Moreover, in 1998, Lortz and Mengenser gave the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order not more than five. Recently, we gave the restricted size Ramsey number for a path of order three and any connected graph of order five. In this paper, we continue the research on the (restricted) size Ramsey number involving small graphs by investigating the restricted size Ramsey number for matching with two edges versus any graph of order five with no isolates

On the restricted size Ramsey number for P3 versus dense connected graphs

Let F, G and H be simple graphs. A graph F is said a (G,H)-arrowing graph if in any red-blue coloring of edges of F we can find a red G or a blue H. The size Ramsey number of G and H, ŕ(G,H), is the minimum size of F. If the order of F equals to the Ramsey number of G and H, r(G,H), then the minimum size of F is called the restricted size Ramsey number of G and H, r*(G,H). The Ramsey number of G and H, r(G,H), is the minimum order of F. In this paper, we study the restricted size number involving a P3.  The value of r*(P3,Kn) has been given by Faudree and Sheehan. Here, we examine r*(P3,H) where H is dense connected graph.</p

Restricted Size Ramsey Number Involving Matching and Graph of Order Five

Harary and Miller (1983) started the research on the (restricted) size Ramsey number for a pair of small graphs. They obtained the values for some pairs of small graphs with order not more than four. In the same year, Faudree and Sheehan continued the research and extended the result to all pairs of small graphs with order not more than four. Moreover, in 1998, Lortz and Mengenser gave the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order not more than five. Recently, we gave the restricted size Ramsey number for a path of order three and any connected graph of order five. In this paper, we continue the research on the (restricted) size Ramsey number involving small graphs by investigating the restricted size Ramsey number for matching with two edges versus any graph of order five with no isolates