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3 Some Results on Maximal Outerplanar Graphs
3.1 A 2-connected Graph Which Is Maximal Outerplanar Graph and Bipartite Is Not Necessarily a Tolerance Graph.
Definition 3.1. A graph G is an outerplanar graph if it has an embedding in the
plane with every vertex on the boundary of the unbounded face. A maximal outerplanar graph is a simple outplanar graph that is not a spanning subgraph of a large simple
out-planar graph.
Figure 37: The left is an example of outerplanar graph and the right is an example of nonouterplanar graph.
In this article, we discuss a graph that is maximal outerplanar and bipartite.
Furthermore, we want to know that whether this graph is a tolerance graph or not.
We adopt that every maximal outerplanar graph in this article is 2-connected.
Figure 38: Some graphs are maximal outerplanar and bipartite.
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國立 政 治 大 學
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N a tio na
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Figure 39: The graph H1. Figure 40: The graph H2.
Figure 41: The graph H3. Figure 42: The graph H4.
Theorem 3.2. Let G be a maximal outerplanar and bipartite graph with vertices number n(G) ≥ 4. G is a tolerance graph if and only if G has no induced subgraphs H1, H2, H3 and H4.
Proof. Let G=(V, E) be a graph that is maximal outerplanar and bipartite. Then we have the following cases of the degree of v where v ∈ V .
Case1: Suppose that the degree of v is not more than 3 for all v ∈ V .
In this case, it is easy to know that G is AT-free. Therefore, by Theorem 2.14, we obtain that G is a bounded tolerance graph. The graph G is shown in Fig-ure 43.
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Figure 43: The graph is maximal outerplanar and bipartite with deg(v) ≤ 3 for all v ∈ V .
Case2: Suppose that there is the only one vertex u∈ V whose degree is greater than 3.
Let u be the common neighbor of v1, v2, ..., vk. Let t1, t2, ..., tk−1 be vertices between vi and vi+1, i = 1, 2, ..., k − 1, respectively. The graph G is shown in Figure 44. We have a tolerance representation of G that is shown in Figure 45. Hence, G is a tolerance graph with tolerances tu = ∞, tvi = 1, for all i = 1, 2, ..., k and ttj = ∞, for all j = 1, 2, ..., k − 1.
Figure 44: The graph is maximal outerplanar and bipartite with deg(u)≥ 4. for the only vertex u ∈ V .
Figure 45: The tolerance representation of G
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Case3: Suppose that there are more than two vertices whose degree are greater than 3.
Let u1 and u2 be two vertices of G with deg( u1)≥ 4 and deg( u2)≥ 4. We have the following relation in u1 and u2.
(i) If the edge u1u2 ∈ E and u1u2 on the boundary of the unbounded face.
For a contradiction, suppose that G is a tolerance graph. Therefore, G has a tolerance representation < I, t >, where I = {Ix|x ∈ V }, t = {tx|x ∈ V }, kxy = |Ix ∩ Iy|, x, y ∈ V . We consider the subgraph H1 of G that is shown in Figure 47 and Figure 48. Because G is a tolerance graph, the subgraph H1 of G also has a tolerance representation. By Lemma 2.16, we know that one of kt2u1 and kv2v3 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G. By Proposition 2.19, we know that kt2u1 is not the smallest value of kxy, for all x, y ∈ {u1, v2, t2, v3} in G. Hence, we get that kv2v3 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G. By Proposition 2.18, we can obtain that kv1u2
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by Proposition 2.19. We reach a contradiction. Therefore, G is not a tolerance graph.Figure 47: The subgraph H1 of G that kt2u1 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G.
Figure 48: The subgraph H1of G that kv2v3 is the smallest value of kxys, for
Similarly, we consider the subgraph H2of G that is shown in the following figures. By Proposition 2.18 and Proposition 2.19, we also prove that G is not a tolerance graph.
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Figure 50: The subgraph H2 of G that kt2u1 is the smallest value of kxy, for all x, y ∈ {u1, v1, t2, v2} in G.
Figure 51: The subgraph H2of G that kv1v2 is the smallest value of kxy, for all x, y ∈ {u1, v1, t2, v2} in G.
(iii) If the edge u1u2 ∈ E and the shortest path from u/ 1 to u2 with odd length is on the boundary of the unbounded face.
Let u1 be the common neighbor of v1, v2, ..., vk, and u2 be the common graph G is shown in Figure 52.
Figure 52: The graph G.
Similarly, we consider the subgraph H3of G that is shown in the following figures. By Proposition 2.18 and Proposition 2.19, we also prove that G is not a tolerance graph.
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Figure 53: The subgraph H3 of G that kt2u1 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G.
Figure 54: The subgraph H3of G that kv2v3 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G.
(iv) If the edge u1u2 ∈ E and the shortest path from u/ 1 to u2 with even length is on the boundary of the unbounded face.
Let u1 be the common neighbor of v1, v2, ..., vk, and u2 be the
The graph G is shown in Figure 55.
Figure 55: The graph G.
We have a tolerance representation of G that is shown in Figure 56.
Hence, G is a tolerance graph with tolerances tu1 = ∞, tu2 = ∞, tvj = 1, for all j = 1, 2, ..., k, tsj = 1, for all j = 1, 2, ..., i,
ttj = ∞, for all j = 1, 2, ..., k − 1, trj = ∞, for all j = 1, 2, ..., i − 1, tpj = 1, for all j = 1, 3, ..., 2n + 1, tpj = ∞, for all j = 2, 4, ..., 2n, tqj = 1, for all j = 2, 4, ..., 2n and tpj = ∞, for all j = 1, 3, ..., 2n + 1.
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Figure 56: The tolerance representation of graph G.
(v) If the edge u1u2 ∈ E and the shortest path from u/ 1 to u2 with odd length is not on the boundary of the unbounded face.
Let u1 be the common neighbor of v1, v2, ..., vk, and u2 be the common neighbor of s1, s2, ..., si. Let t1, t2, ..., tk−1 be vertices between vi−1 and vi, i = 1, 2, ..., k, respectively and r1, r2, ..., ri−1be vertices between sj−1 and sj, j = 1, 2, ..., i, respectively. Let p1, p2, ..., p2n+1 be vertices of the shortest path from u1 to s1 and q1, q2, ..., q2n+1 be vertices of the shortest path from v1 to u2 where pjqj ∈ E, for all j = 1, ..., 2n + 1. The graph G is shown in Figure 57.
Figure 57: The graph G.
Similarly, we consider the subgraph H4 of G that is shown in the fol-lowing figures. By Proposition 2.18 and Proposition 2.19, we also prove that G is not a tolerance graph.
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Figure 58: The subgraph H4 of G that kt2u1 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G.
Figure 59: The subgraph H4of G that kv2v3 is the smallest value of kxys, for all x, y ∈ {u1, v2, t2, v3} in G.
(vi) If the edge u1u2 ∈ E and the shortest path from u/ 1 to u2 with even length is not on the boundary of the unbounded face.
Let u1 be the common neighbor of v1, v2, ..., vk and u2 be the common
The graph G is shown in Figure 60.
Figure 60: The graph G.
We have a tolerance representation of G as Figure 61 shown. Hence, G is a tolerance graph with tolerances tu1 = ∞, tu2 = ∞,
tvj = 1, for all i = 1, 2, ..., k , tsj = 1, for all j = 1, 2, ..., i − 1, ttj = ∞, for all j = 1, 2, ..., k − 1, trj = ∞, for all j = 1, 2, ..., i − 1,
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tpj = 1, for all j = 1, 3, ..., 2n − 1, tpj = ∞, for all j = 2, 4, ..., 2n, tqj = 1, for all j = 2, 4, ..., 2n and tpj = ∞, for all j = 1, 3, ..., 2n − 1.
Figure 61: The tolerance representation of graph G.