Plane graphs of diameter two are 2-optimal
Introduction
Let be a positive integer. Assume a fire starts at some vertex of a connected graph . A firefighter (or defender) protect 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 be -vertex graph. The damage number for a vertex , denoted by , is the minimum number of vertices in that the fire may burn if it begins at vertex . The average proportion of vertices that can be damaged from the fire if it begins randomly in is known as the -expected damage of a graph , that is,Another way of looking at the expected damage is the surviving rate, denoted by , due to Cai and Wang [2]. In particular, setting .
Obviously, . It is proved that almost all graphs have arbitrarily close to 1 by Wang et al.[9]. For a positive constant , we say that a class of graphs with vertices is -good if for any . Additionally, is -optimal if for any . The primary purpose of Firefighter Problem is to determine whether is -good or -optimal for a given number .
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 , it is conjectured that planar graphs are 2-good. The girth is the length of a shortest cycle of a graph . In particular, the girth of tree is infinite. Planar graphs with girth at least 7 are 1-good [10]. Since is a planar graph of girth 4 and , 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 are -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 and be two vertices of . The distance between and is the length of a shortest path connecting them. The diameter of a graph is the maximum distance between any two vertices of . 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 MPD-graph. Recently, Cui et al.[3] investigated the structure and pancyclicity of MPD-graphs.
The planar graphs do not have bounded treewidth. However, Eppstein [4] confirmed that planar graphs with diameter have treewidth at most . This implies that MPD-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 of diameter two, as shown in Fig. 1, which satisfies .
As an easy observation, for any spanning subgraph of a graph , we have . So, among all planar graphs of diameter two, MPD-graphs attain the largest expected damage. Hence, it suffices to consider the 2-expected damage of MPD-graphs in the following argument.
Section snippets
Preliminaries
Let be a plane graph. We denote the set of faces of by . A cycle of a plane graph is said to be separating if both its interior and exterior contain at least one vertex of , and the set of vertices in the interior of are referred to as . Let and be two vertex-disjoint graphs, the join of and , denoted , is obtained by adding edges joining every vertex of to every vertex of .
Let be an MPD-graph and be a 3-cycle in . There are three configurations,
Preliminary results
Let be an integer. A vertex is manageable if dn. We use to denote an ordered defending strategy when a fire breaks out at in , where the -th term is the set of vertices that we need to protect in time interval . Let denote the number of faces, which admit feasible operations, incident to , and let . Lemma 3.1 Let be an MPD-graph with . Then every vertex in is manageable of . Proof By Theorem 2,
Main results
According to Lemmas 3.2-3.9, every MPD-graph with has . Thus we have the following theorem. Theorem 3 If is an MPD-graph with , then . Theorem 4 If is an outerplanar graph of order , then .[1]
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 MPD-graph with and a maximal outerplanar graph. Theorem 5 If is
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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).