Plane graphs of diameter two are 2-optimal

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Highlights

  • A new class of graphs which is k-optimal is presented.

  • For planar graphs of diameter two, 2-optimal cannot be improved.

  • It can be applied to network design and the propagation of rumours, viruses or epidemics in networks.

Abstract

Let G be a connected graph with n vertices. Suppose a fire breaks out at some vertex of G. At each time interval, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. Let dnk(v) denote the minimum number of vertices that the fire may damage when a fire breaks out at vertex v. The k-expected damage of G, denoted by εk(G), is the expectation of the proportion of vertices that can be damaged from the fire, if the starting vertex of the fire is chosen uniformly at random, i.e., εk(G)=vV(G)dnk(v)/n2. A class of graphs G is called k-optimal if εk(G) tends to 0 as n tends to infinity for any GG. In this paper, we prove that planar graphs of diameter two are 2-optimal, which is the best possible.

Introduction

Let k be a positive integer. Assume a fire starts at some vertex of a connected graph G. A firefighter (or defender) protect k non-burning vertices. The firefighter and the fire alternate on the graph. When a firefighter selects a vertex, it is regarded protected from further fire movement. After the firefighter moves, the fire spreads to all adjacent vertices, except those protected. When the fire is no longer able to spread, the process comes to an end. This is a simple deterministic model of the spread of fire, illnesses and computer viruses in graphs, known as the “Firefighter Problem”.

Let G be n-vertex graph. The damage number for a vertex v, denoted by dnk(v), is the minimum number of vertices in G that the fire may burn if it begins at vertex v. The average proportion of vertices that can be damaged from the fire if it begins randomly in G is known as the k-expected damage εk(G) of a graph G, that is,εk(G)=vV(G)dnk(v)n2.Another way of looking at the expected damage is the surviving rate, denoted by ρk(G)=1εk(G), due to Cai and Wang [2]. In particular, setting ε1(G)=ε(G).

Obviously, 0<εk(G)<1. It is proved that almost all graphs have εk(G) arbitrarily close to 1 by Wang et al.[9]. For a positive constant c, we say that a class of graphs G with n vertices is k-good if εk(G)<c<1 for any GG. Additionally, G is k-optimal if limnεk(G)=0 for any GG. The primary purpose of Firefighter Problem is to determine whether G is k-good or k-optimal for a given number k.

A planar graph is a graph that can be drawn on the plane in such a way that no edges cross each other. A specific drawing of a planar graph is called plane graph. The surviving rate of planar graphs has been studied intensively in the last decade. Planar graphs are 5-good [9]; 4-good [5] and [7]; 3-good [6] and [8]. Since limnε(K2,n)=1, it is conjectured that planar graphs are 2-good. The girth is the length of a shortest cycle of a graph G. In particular, the girth of tree is infinite. Planar graphs with girth at least 7 are 1-good [10]. Since K2,n is a planar graph of girth 4 and limnε(K2,n)=1, it is conjectured that planar graphs of girth at least 5 are 1-good. It is proved that trees are 1-optimal [2], graphs with treewidth k are k-optimal [1], and outerplanar graphs are 1-optimal [1]. We refer the reader to [11] for a newest survey on the surviving rate of graphs.

Let u and v be two vertices of G. The distance between u and v is the length of a shortest path connecting them. The diameter of a graph G is the maximum distance between any two vertices of G. A simple graph is called maximal planar if it is planar but adding any edge (on the given vertex set) would destroy that property. A maximal plane graph with diameter two is simply said to be an MPD2-graph. Recently, Cui et al.[3] investigated the structure and pancyclicity of MPD2-graphs.

The planar graphs do not have bounded treewidth. However, Eppstein [4] confirmed that planar graphs with diameter d have treewidth at most 3d2. This implies that MPD2-graphs are 4-optimal. The main purpose of this paper is to prove the following:

Theorem 1

Every planar graph of diameter two is 2-optimal.

Theorem 1 is best possible, since there exists a non-1-optimal maximal planar graph H of diameter two, as shown in Fig. 1, which satisfies ε(H)=n2n=1(n).

As an easy observation, for any spanning subgraph H of a graph G, we have εk(H)εk(G). So, among all planar graphs of diameter two, MPD2-graphs attain the largest expected damage. Hence, it suffices to consider the 2-expected damage of MPD2-graphs in the following argument.

Section snippets

Preliminaries

Let G be a plane graph. We denote the set of faces of G by F(G). A cycle C of a plane graph G is said to be separating if both its interior and exterior contain at least one vertex of G, and the set of vertices in the interior of C are referred to as Int(C). Let G and H be two vertex-disjoint graphs, the join of G and H, denoted GH, is obtained by adding edges joining every vertex of G to every vertex of H.

Let G be an MPD2-graph and T=[xyz] be a 3-cycle in G. There are three configurations,

Preliminary results

Let c1 be an integer. A vertex is manageable if dn2(v)c. We use Γ(v,G)={γ1,γ2,,γm} to denote an ordered defending strategy when a fire breaks out at v in G, where the t-th term γt is the set of vertices that we need to protect in time interval t1. Let β(v) denote the number of faces, which admit feasible operations, incident to v, and let β(G)=vV(G)β(v).

Lemma 3.1

Let G be an MPD2-graph with Δ(G)|G|2. Then every vertex in V(Tz*)V(Tzx*)V(Txyz*)V(Qwz*) is manageable of G.

Proof

By Theorem 2, Gi=110F(Gi

Main results

According to Lemmas 3.2-3.9, every MPD2-graph G with Δ(G)n2 has dn2(G)17n. Thus we have the following theorem.

Theorem 3

If G is an MPD2-graph with Δ(G)n2, then ε2(G)=O(1n).

Theorem 4

[1]

If G is an outerplanar graph of order n, then ε(G)=O(lognn).

A maximal outerplanar graph is an outerplanar graph that cannot have any additional edges added to it while preserving outerplanarity. Cui et al. [3] verified that there exists a one-to-one mapping between MPD2-graph with Δ(G)=n1 and a maximal outerplanar graph.

Theorem 5

If G is

References (11)

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Cited by (0)

1

Research supported by China Postdoctoral Science Foundation (No. 2020M681927) and Fundamental Research Funds for the Provincial Universities of Zhejiang (No. 2021YW08).

2

Research supported by NSFC (Nos. 12071048; 12161141006).

3

Research supported by NSFC (Nos. 12031018; 12226303).

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