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Embedding Cycle Graphs Complements Liliek Susilowati; Hendy Hendy; Yayuk Wayuni
Jurnal ILMU DASAR Vol 9 No 2 (2008)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (218.616 KB)

A graph is embeddable on a surface if it can be drawn on that surface without any edges intersect. The cycle graphs can always be embedded on the plane and the torus, but this is not occurred for their complements. We prove that the maximum order of cycle graphs such that their complements still can be embedded on the plane is 6. But, the maximum order of cycle graphs such that their complements still can be embedded on the torus is 9. Also, the crossing number of complements of cycle graphs which can’t be embedded on the plane with minimum order will be presented.

Embedding K2,3,m Graphs on Torus Liliek Susilowati
Jurnal ILMU DASAR Vol 10 No 2 (2009)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

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This script aim to determine the maximal value of m of K2,3,m graph so that can be redrawn on torus without intersection of edges. Hereinafter, will be determined the toroidal crossing number of K2,3,m which nontoroidal by the m minimize. To determined if the K2,3,m graph is toroidal, it is enough with drawn the graph on torus without intersection of edges, whereas, to determined if it is nontoroidal, besides with drawn, is also needed by theorem about properties of graph that containing K5-subdivision. Then to determined the toroidal crossing number was used the technique by proof of the crossing number of K2,2,3 and looked for of all probabilities of the edges is going to intersect. In this research, was obtained the result that maximal value of m of K2,3,m so that can be drawn on torus without intersection of edges is 3, while the toroidal crossing number of K2,3,4 is 2, and our conjecture is tcr().

On Commutative Characterization of Graph Operation with Respect to Metric Dimension Liliek Susilowati; Mohammad Imam Utoyo; Slamin Slamin
Journal of Mathematical and Fundamental Sciences Vol. 49 No. 2 (2017)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2017.49.2.5

Let G be a connected graph with vertex set V(G) and W={w1, w2, ..., wm} âŠ† V(G). A representation of a vertex v âˆˆ V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),...,d(v,wm)) where d(v,w) is the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.

DIMENSI METRIK KETETANGGAAN LOKAL GRAF HASIL OPERASI k-COMB Fryda Arum Pratama; Liliek Susilowati; Moh. Imam Utoyo
Contemporary Mathematics and Applications (ConMathA) Vol. 1 No. 1 (2019)
Publisher : Universitas Airlangga

Research on the local adjacency metric dimension has not been found in all operations of the graph, one of them is comb product graph. The purpose of this research was to determine the local adjacency metric dimension of k-comb product graph and level comb product graph between any connected graph G and H. In this research graph G and graph H such as cycle graph, complete graph, path graph, and star graph. K-comb product graph between any graph G and H denoted by GokH. While level k comb product graph between any graph G and H denoted by GokH.In this research, local adjacency metric dimension of GokSm graph only dependent to multiplication of the cardinality of V(G) and many of k value, while GokKm graph and GokCm graph is dependent to dominating number of G and multiplication of the cardinality of V(G), many of k value, and local adjacency metric dimension of Km graph or Cm graph. And then, local adjacency metric dimension of GokSm graph only dependent to the cardinality of V(Gok-1Sm), while GokKm graph and GokCm graph is dependent to dominating number of G and multiplication of the local adjacency metric dimension of Km graph or Cm graph with cardinality of V(Gok-1Km) or V(Gok-1Cm).

Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian Nurma Ariska Sutardji; Liliek Susilowati; Utami Dyah Purwati
Contemporary Mathematics and Applications (ConMathA) Vol. 1 No. 2 (2019)
Publisher : Universitas Airlangga

The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

Hubungan Dimensi Metrik Ketetanggaan dan Dimensi Metrik Ketetanggan Lokal Graf Hasil Operasi Kali Korona Virdina Rahmayanti; Moh. Imam Utoyo; Liliek Susilowati
Contemporary Mathematics and Applications (ConMathA) Vol. 2 No. 1 (2020)
Publisher : Universitas Airlangga

Adjacency metric dimension and local adjacency metric dimension are the development of metric dimension. The purpose of this research is to determine the adjacency metric dimension of corona graph between any connected graph G and non-trivial graph H denoted by dimA(G⊙H), to determine the local adjacency metric dimension of corona graph between any connected graph G and non-trivial graph H denoted by dimA,l(G⊙H), and to determine the correlation between adjacency metric dimension and local adjacency metric dimension of corona product graph operations. In this research, it is found out that the value of adjacency metric dimension of G⊙H graph is affected by the basic characteristic of H and the domination characteristic. Meanwhile, the value of local adjacency metric dimension of G⊙H graph is only affected by the basic characteristic of H Futhermore, it is found a correlation of adjacency metric dimension and local adjacency metric dimension of corona product graph between any connected graph G and non-trivial graph H.

On the Dominant Local Resolving Set of Vertex Amalgamation Graphs Reni Umilasari; Liliek Susilowati; S Slamin; Savari Prabhu
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.18891

Basically, the new topic of the dominant local metric dimension which be symbolized by Ddim_l (H) is a combination of two concepts in graph theory, they were called the local metric dimension and dominating set. There are some terms in this topic that is dominant local resolving set and dominant local basis. An ordered subset W_l is said a dominant local resolving set of G if W_l is dominating set and also local resolving set of G. While dominant local basis is a dominant local resolving set with minimum cardinality. This study uses literature study method by observing the local metric dimension and dominating number before detecting the dominant local metric dimension of the graphs. After obtaining some new results, the purpose of this research is how the dominant local metric dimension of vertex amalgamation product graphs. Some special graphs that be used are star, friendship, complete graph and complete bipartite graph. Based on all observation results, it can be said that the dominant local metric dimension for any vertex amalgamation product graph depends on the dominant local metric dimension of the copied graphs and how the terminal vertex is constructed

On Rainbow Antimagic Coloring of Joint Product of Graphs Brian Juned Septory; Liliek Susilowati; Dafik Dafik; M. Venkatachalam
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.17471

Let be a connected graph with vertex set and edge set . A bijection from to the set is a labeling of graph . The bijection is called rainbow antimagic vertex labeling if for any two edge and in path , where and . Rainbow antimagic coloring is a graph which has a rainbow antimagic labeling. Thus, every rainbow antimagic labeling induces a rainbow coloring G where the edge weight is the color of the edge . The rainbow antimagic connection number of graph is the smallest number of colors of all rainbow antimagic colorings of graph , denoted by . In this study, we studied rainbow antimagic coloring and have an exact value of rainbow antimagic connection number of joint product of graph where is graph , graph , graph , graph and graph .

On the Dominant Local Resolving Set of Vertex Amalgamation Graphs Umilasari, Reni; Susilowati, Liliek; Slamin, S; Prabhu, Savari
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.18891

Basically, the new topic of the dominant local metric dimension which be symbolized by Ddim_l (H) is a combination of two concepts in graph theory, they were called the local metric dimension and dominating set. There are some terms in this topic that is dominant local resolving set and dominant local basis. An ordered subset W_l is said a dominant local resolving set of G if W_l is dominating set and also local resolving set of G. While dominant local basis is a dominant local resolving set with minimum cardinality. This study uses literature study method by observing the local metric dimension and dominating number before detecting the dominant local metric dimension of the graphs. After obtaining some new results, the purpose of this research is how the dominant local metric dimension of vertex amalgamation product graphs. Some special graphs that be used are star, friendship, complete graph and complete bipartite graph. Based on all observation results, it can be said that the dominant local metric dimension for any vertex amalgamation product graph depends on the dominant local metric dimension of the copied graphs and how the terminal vertex is constructed

On Rainbow Antimagic Coloring of Joint Product of Graphs Septory, Brian Juned; Susilowati, Liliek; Dafik, Dafik; Venkatachalam, M.
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.17471

Let be a connected graph with vertex set and edge set . A bijection from to the set is a labeling of graph . The bijection is called rainbow antimagic vertex labeling if for any two edge and in path , where and . Rainbow antimagic coloring is a graph which has a rainbow antimagic labeling. Thus, every rainbow antimagic labeling induces a rainbow coloring G where the edge weight is the color of the edge . The rainbow antimagic connection number of graph is the smallest number of colors of all rainbow antimagic colorings of graph , denoted by . In this study, we studied rainbow antimagic coloring and have an exact value of rainbow antimagic connection number of joint product of graph where is graph , graph , graph , graph and graph .

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