PAM 271 PENGANTAR TEORI GRAF

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1 PAM 271 PENGANTAR TEORI GRAF SEMESTER GANJIL Lyra Yulianti Jurusan Matematika FMIPA Universitas Andalas LYRA (MA-UNAND) 1 / 15

2 Outline Outline 1 Kontrak Kuliah LYRA (MA-UNAND) 2 / 15

3 Outline Outline 1 Kontrak Kuliah 2 Graphs and Subgraphs Graphs Isomorphism The Incidence and Adjacency Matrix LYRA (MA-UNAND) 2 / 15

4 Kontrak Kuliah Jadwal dan Buku Pegangan Jadwal Tatap Muka : Kamis, (3 SKS), Ruang B.1.7 Buku Pegangan Wajib: Bondy, J.A., and Murty, U.S.R., Graph Theory with Applications, The Macmillan Press LTD, Great Britain, 1976 Pendukung: Hartsfield, N., and Ringel G., Pearls in Graph Theory: A Comprehensive Introduction, 2 nd Edition, Academic Press, New York, 1990 LYRA (MA-UNAND) 3 / 15

5 Kontrak Kuliah Penilaian No Komponen Penilaian Persentase Penilaian Hasil 1 Ujian Tengah Semester 30 2 Ujian Akhir Semester 30 3 Kuis 10 Penilaian Proses 1 Kemampuan berpikir kritis dan berargumen logis 15 2 Kerjasama dalam tim 15 Total 100 LYRA (MA-UNAND) 4 / 15

6 Kontrak Kuliah Materi Perkuliahan Pokok Bahasan Graf dan subgraf, graf pohon, keterhubungan (connectivity), tour Euler dan lingkaran Hamilton, matching, pewarnaan titik, pewarnaan sisi, bilangan kromatik, graf planar, teorema Kuratowski, serta beberapa aplikasi sederhana. LYRA (MA-UNAND) 5 / 15

7 Kontrak Kuliah Materi Perkuliahan Pokok Bahasan Graf dan subgraf, graf pohon, keterhubungan (connectivity), tour Euler dan lingkaran Hamilton, matching, pewarnaan titik, pewarnaan sisi, bilangan kromatik, graf planar, teorema Kuratowski, serta beberapa aplikasi sederhana. Capaian Pembelajaran Setelah mengikuti perkuliahan ini, mahasiswa diharapkan mampu: (a) Memahami, menerangkan serta menggunakan beberapa konsep sederhana yang menjadi dasar dalam teori graf. (b) Mengidentifikasi hubungan antara masalah-masalah dalam matakuliah ini dengan cabang matematika yang lain, begitu juga dengan cabang-cabang ilmu yang lainnya. (c) Berpikir kritis, analitis dan inovatif, dapat berargumen secara logis dan terstruktur. (d) Mengkomunikasikan buah pikiran secara sistematis, dapat bekerja sama dan beradaptasi dengan mahasiswa lain dalam kelompok, serta melakukan diskusi dengan baik. LYRA (MA-UNAND) 5 / 15

8 Outline 1 Kontrak Kuliah 2 Graphs and Subgraphs Graphs Isomorphism The Incidence and Adjacency Matrix LYRA (MA-UNAND) 6 / 15

9 The Königsberg Bridge Problem In 1736, the city of Königsberg was located in Prussia (in Europe). The River Pregel flowed through the city dividing it into four land areas, namely 1, 2, 3 and 4. Seven bridges were built across the river, namely a, b,, g. Figure: The Königsberg Bridge LYRA (MA-UNAND) 7 / 15

10 The Famous Königsberg Bridge Problem Question Is it possible to walk about Königsberg crossing each of its seven bridges exactly once? Figure: The Graph Representation of Königsberg Bridge Problem LYRA (MA-UNAND) 8 / 15

11 The Famous Königsberg Bridge Problem Question Is it possible to walk about Königsberg crossing each of its seven bridges exactly once? Figure: The Graph Representation of Königsberg Bridge Problem Answer No. Why? (GROUP ASSIGNMENT NO.1) LYRA (MA-UNAND) 8 / 15

12 The Three Friends or Three Strangers Problem Question What is the smallest number of people that must be present at a gathering to be certain that among them, three are mutual friends or three are mutual strangers? LYRA (MA-UNAND) 9 / 15

13 The Three Friends or Three Strangers Problem Question What is the smallest number of people that must be present at a gathering to be certain that among them, three are mutual friends or three are mutual strangers? Answer Six. Why? LYRA (MA-UNAND) 9 / 15

14 The Three Friends or Three Strangers Problem Question What is the smallest number of people that must be present at a gathering to be certain that among them, three are mutual friends or three are mutual strangers? Answer Six. Why? Hint Ramsey Theory LYRA (MA-UNAND) 9 / 15

15 Some Graph Definitions A graph G = (V, E, ψ G ) is an ordered triple, consists of a nonempty set V (G) of vertices, a set E(G) of edges, and an incidence function ψ G that associates with each edge of G an ordered pair of (not necessarily distinct) vertices of G. If e is an edge and u and v are vertices such that ψ G (e) = uv, then e is said to join u and v, the vertices u and v are called the ends of e. The ends of an edge are said to be incident with the edge, and vice versa. Two vertices which are incident with a common edge are adjacent. An edge with identical ends is called a loop, an edge with distinct ends a link. A graph is finite if both its vertex set and edge set are finite. A graph with just one vertex trivial and all other graphs nontrivial. A graph is simple if it has no loops and no two of its link join the same pair of vertices. LYRA (MA-UNAND) 10 / 15

16 Some Graph Definitions Let G = (V, E). The number of vertices is denoted by V (G) = p and the number of edges is denoted by E(G) = q. Determine V (G) and E(G). Figure: Graph G Show that if G is simple then q ( p 2). LYRA (MA-UNAND) 11 / 15

17 Graphs Isomorphism Outline 1 Kontrak Kuliah 2 Graphs and Subgraphs Graphs Isomorphism The Incidence and Adjacency Matrix LYRA (MA-UNAND) 12 / 15

18 Graphs Isomorphism Some Definitions Two graphs G and H are identical, denoted by G = H, if V (G) = V (H), E(G) = E(H) dan ψ G = ψ H. Two graphs G and H are isomorphic, denoted by G = H if there are bijection θ : V (G) V (H) and φ : E(G) E(H) such that ψ G (e) = uv if and only if ψ H (φ(e)) = θ(u)θ(v); such a pair θ, φ of mappings is called an isomorphism between G and H. Figure: Isomorphisms Show that those graphs are isomorphic (GROUP ASSIGNMENT NO.2) Show that there are eleven non-isomorphic simple graphs on four vertices. (GROUP ASSIGNMENT NO.3) LYRA (MA-UNAND) 13 / 15

19 The Incidence and Adjacency Matrix Outline 1 Kontrak Kuliah 2 Graphs and Subgraphs Graphs Isomorphism The Incidence and Adjacency Matrix LYRA (MA-UNAND) 14 / 15

20 The Incidence and Adjacency Matrix The Incidence Matrix Figure: Incidence matrices LYRA (MA-UNAND) 15 / 15