A note on domination 3-edge-critical planar graphs Michitaka Furuya∗and Naoki Matsumoto†

Abstract For a graph G, we let γ(G) denote the domination number of G. A graph G is 3-edge-critical if γ(G) = 3 and γ(G + xy) ≤ 2 for all x, y ∈ V (G) with xy ∈ / E(G). In this note, we show that the order of a connected 3-edge-critical planar graph is at most 26.

Key words and phrases. domination 3-edge-critical graph, planar graph, factorbicritical graph. AMS 2010 Mathematics Subject Classification. 05C69, 05C10.

1

Introduction

All graphs considered in this paper are finite, simple, and undirected. Let G be a graph. We let V (G) and E(G) denote the vertex set and the edge set of G, respectively. For x ∈ V (G), let NG (x), NG [x] and dG (x) denote the open neighborhood, the closed neighborhood and the degree of x, respectively; thus NG (x) = {y ∈ V (G) : xy ∈ E(G)}, NG [x] = NG (x) ∪ {x} and dG (x) = |NG (x)|. Let ∆(G) denote the maximum degree of G. For X ⊆ V (G), we let G[X] denote the subgraph of G induced by X. We let Kn and Km1 ,m2 denote the complete graph of order n and the complete bipartite graph with partite sets having cardinalities m1 and m2 , respectively. A subset X of V (G) is a dominating set of G if V (G) =

∪ x∈X

NG [x]. The mini-

mum cardinality of a dominating set of G, denoted by γ(G), is called the domination number of G. A dominating set of G with the cardinality γ(G) is called a γ-set of G. For an integer k ≥ 1, a graph G is k-edge-critical if γ(G) = k and γ(G + xy) ≤ k − 1 for all x, y ∈ V (G) with xy ∈ / E(G). Note that a graph is 1-edge-critical if and only if the graph is a complete graph. Furthermore, Sumner and Blitch [5] proved that ∗

College of Liberal Arts and Science, Kitasato University, 1-15-1 Kitasato, Minami-ku, Sagami-

hara, Kanagawa 252-0373, Japan, E-mail: [email protected] † Department of Computer and Information Science, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino-shi, Tokyo 180-8633, Japan, E-mail: [email protected]

1

+

+

Kn

K2

Figure 1: Graph in Lemma 2.2 (where the symbol “+” denotes the join of vertices)

a graph is 2-edge-critical if and only if each component of its complement is a star. Thus we are interested in k-edge-critical graphs for k ≥ 3. Among them, 3-edgecritical graphs have been especially studied by many researchers (see, for example, [3, 4, 6, 7]). Ananchuen and Plummer [2] studied the factor-criticality in 3-edge-critical graphs. They gave the following theorem as one of corollaries of their main result. Theorem A (Ananchuen and Plummer [2]) If G is a 3-connected 3-edge-critical planar graph having even order, then G is factor-bicritical. Here one question might be naturally posed: How large is the class of 3-edgecritical planar graphs? In this note, we give an answer for the question showing that the class is very small. Theorem 1.1 For a connected 3-edge-critical planar graph G, |V (G)| ≤ 26.

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Proof of Theorem 1.1

In our proof, we will use the following lemmas. Lemma 2.1 (Sumner and Blitch [5]) Let G be a connected 3-edge-critical graph. Then |{x ∈ V (G) : dG (x) ≤ i}| ≤ 3i for each i ≥ 1. Lemma 2.2 (Ananchuen and Plummer [1]) Let G be a connected 3-edge-critical graph. If G has at least two vertices of degree exactly 1, then G is the graph depicted in Figure 1 for m ≥ 1. Proof of Theorem 1.1.

Let G be a connected 3-edge-critical planar graph of order

n. By way of contradiction, suppose that n ≥ 27. For each i ≥ 1, let Vi = {x ∈ V (G) : dG (x) = i}, and set ni = |Vi |, n− i = ∑ ∪ + − + − and ni = j≥i nj . Let S = 1≤i≤6 Vi and S = V (G) − S .

2

∑

j≤i nj

Claim 2.1 Let h (2 ≤ h ≤ 5) and t ≥ 0. If n− h ≤ t, then ∑

11 ≤

∑

min{t, 3i} +

2≤i≤h

3i −

∑

(i − 6)ni .

i≥7

h+1≤i≤5

Proof. Since G contains no K3,3 , it follows from Lemma 2.2 that if n1 ≥ 2, then n ≤ 7, which is a contradiction. Thus we have n− 1 = n1 ≤ 1. Hence by Lemma 2.1,   ∑ ∑ ∑ ∑ ∑ ∑  (6 − i)ni = nj  = n− 1 + min{t, + 3i. ≤ 3i} i 1≤i≤5

1≤i≤5

1≤j≤i

1≤i≤5

2≤i≤h

h+1≤i≤5

This together with Euler’s formula leads to 12 ≤

∑

(6 − i)ni ≤ 1 +

i≥1

as desired.

∑

min{t, 3i} +

2≤i≤h

∑ h+1≤i≤5

3i −

∑ (i − 6)ni , i≥7

□

If n− 6 ≤ 4, then by applying Claim 2.1 for h = 5 and t = 4, we have 11 ≤ (4 + 4 + 4 + 4) −

∑ (i − 6)ni ≤ 16 − n+ 7, i≥7

− + and hence n = n− 6 + n7 ≤ 4 + 5 = 9, which is a contradiction. Thus n6 ≥ 5.

Claim 2.2 Let x, y ∈ S − with xy ∈ / E(G), and let D be a γ-set of G + xy. Then |D| = 2 and D contains exactly one of x and y. Furthermore, if x ∈ D, then uy ∈ / E(G), dG (u) ≥ n − dG (x) − 3 and u ∈ S + where u is the unique vertex in D − {x}. Proof. By the definition of the 3-edge-criticality, it suffices to only show that dG (u) ≥ n − dG (x) − 3 and u ∈ S + . Since D = {x, u} is a dominating set of G + xy, V (G) = NG+xy [x] ∪ NG+xy [u], and so n = |V (G)| ≤ |NG+xy [x]| + |NG+xy [u]| = (dG (x) + 2) + |NG [u]|. This implies that dG (u) = |NG [u]| − 1 ≥ n − dG (x) − 3. If u ∈ S − , then 6 ≥ dG (u) ≥ n − dG (x) − 3 ≥ n − 9, which is a contradiction.

□

− − Since n− 6 ≥ 5 and G[S ] contains no K5 , there exist two vertices x1 , y1 ∈ S

with x1 y1 ∈ / E(G). We choose x1 and y1 so that max{dG (x1 ), dG (y1 )} is as small as possible. Let D1 be a γ-set of G + x1 y1 . Without loss of generality, we may assume that x1 ∈ D1 . By Claim 2.2, we can write D1 = {x1 , u1 } with u1 ̸= y1 , and we have u1 y1 ∈ / E(G), dG (u1 ) ≥ n − dG (x1 ) − 3 and u1 ∈ S + . Suppose that n− 4 ≤ 4. Then by applying Claim 2.1 for h = 4 and t = 4, we have 11 ≤ (4 + 4 + 4 + 15) −

∑

(i − 6)ni ≤ 27 − (n+ 7 − 1) − ((n − dG (x1 ) − 3) − 6),

i≥7

3

and hence n+ 7 ≤ 26 + dG (x1 ) − n ≤ 32 − n. This together with Lemma 2.1 leads + to n = n− 6 + n7 ≤ 18 + (32 − n), and so n ≤ 25, which is a contradiction. Thus ∪ n− 4 ≥ 5. Since G[ 1≤i≤4 Vi ] contains no K5 , it follows from the choice of x1 and y1 ∪ that x1 , y1 ∈ 1≤i≤4 Vi . In particular, dG (x1 ) ≤ 4.

Suppose that n− 6 ≤ 11. Then by applying Claim 2.1 for h = 5 and t = 11, we have 11 ≤ (6 + 9 + 11 + 11) −

∑ (i − 6)ni ≤ 37 − (n+ 7 − 1) − ((n − dG (x1 ) − 3) − 6), i≥7

and hence n+ 7 ≤ 36 + dG (x1 ) − n ≤ 40 − n. This together with Lemma 2.1 leads + to n = n− 6 + n7 ≤ 11 + (40 − n), and so n ≤ 25, which is a contradiction. Thus

n− 6 ≥ 12. Set U = NG [u1 ] ∩ S − . Since D1 is a dominating set of G + x1 y1 and dG (x1 ) ≤ 4 + (i.e., |NG+x1 y1 [x1 ]| ≤ 6), |U | ≥ |S − | − 6. Since n− 6 ≥ 12 and u1 ∈ S , it follows that

|U | ≥ |S − | − 6 ≥ 6. Since G[U ] contains no K5 , there exist two vertices x2 , y2 ∈ U with x2 y2 ∈ / E(G). Let D2 be a γ-set of G + x2 y2 . Without loss of generality, we may assume that x2 ∈ D2 . By Claim 2.2, we can write D = {x2 , u2 } with u2 ̸= y2 , and we have u2 y2 ∈ / E(G), dG (u2 ) ≥ n − dG (x2 ) − 3 ≥ n − 9 and u2 ∈ S + . Since u1 is adjacent to y2 in G, u2 ̸= u1 . Recall that dG (u1 ) ≥ n − dG (x1 ) − 3 ≥ n − 7. Hence by applying Claim 2.1 for h = 5 and t = ∞, we have ∑ 11 ≤ (6 + 9 + 12 + 15) − (i − 6)ni ≤ 42 − (n+ 7 − 2) − ((n − 7) − 6) − ((n − 9) − 6), i≥7 + − and hence n+ 7 ≤ 61 − 2n. This together with Lemma 2.1 leads to n = n6 + n7 ≤

18 + (61 − 2n), and so n ≤ 26, which is a contradiction. This completes the proof of Theorem 1.1.

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□

Concluding remark

In this note, we proved that if G is a connected 3-edge-critical planar graph, then |V (G)| ≤ 26. Since our aim is to show that the class of 3-edge-critical planar graphs is very small, we have not referred to the sharpness of the bound. Indeed, we expect that the bound can be improved. However, it seems a time-consuming work as long as we use similar argument in this note even if we slightly improve the bound. Thus we leave the problem to give a sharp bound or to characterize the class of 3-edge-critical planar graphs for the readers. One might be interested in a more general class, that is, the class of 3-edgecritical graphs on a closed surface F 2 . We conclude this paper by showing the following theorem. 4

Theorem 3.1 Let F 2 be a closed surface. Then there exists a constant c = c(F 2 ) such that every connected 3-edge-critical graph G on F 2 satisfies |V (G)| ≤ c. Proof. We show that for every connected 3-edge-critical graph G on F 2 , |V (G)| ≤ (51 − 6ϵ(F 2 ))3 where ϵ(F 2 ) is the Euler characteristic of F 2 . It is known that the diameter of a connected 3-edge-critical graph is at most 3 (see [5]). Since |V (G)| ≤ (∆(G))d where d is the diameter of G, it suffices to show that ∆(G) ≤ 51 − 6ϵ(F 2 ). For each i, we similarly define Vi , ni and n− i as in the proof of Theorem 1.1. Then by Lemma 2.1, ∑

(6 − i)ni =

1≤i≤5

∑ 1≤i≤5

 

∑ 1≤j≤i



∑

nj  =

n− i ≤ 3 + 6 + 9 + 12 + 15 = 45.

1≤i≤5

This together with Euler’s formula leads to 6ϵ(F 2 ) ≤

∑ ∑ (6 − i)ni ≤ 45 − (i − 6)ni ≤ 45 − (∆(G) − 6), i≥1

i≥7

and hence ∆(G) ≤ 51 − 6ϵ(F 2 ), as desired.

□

Acknowledgment This work was supported by JSPS KAKENHI Grant number 26800086 (to M.F).

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[7] E. Wojcicka, Hamiltonian properties of domination-critical graphs, J. Graph Theory 14 (1990) 205–215.

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