aa r X i v : . [ m a t h . C O ] N ov PLANAR GRAPH IS ON FIRE
PRZEMYSŁAW GORDINOWICZ
Abstract.
Let G be any connected graph on n vertices, n ≥ . Let k be any positiveinteger. Suppose that a fire breaks out at some vertex of G. Then, in each turn k firefighters can protect vertices of G — each can protect one vertex not yet on fire;Next the fire spreads to all unprotected neighbours.The k -surviving rate of G, denoted by ρ k ( G ) , is the expected fraction of verticesthat can be saved from the fire by k firefighters, provided that the starting vertex ischosen uniformly at random. In this paper, it is shown that for any planar graph G wehave ρ ( G ) ≥ . Moreover, 3 firefighters are needed for the first step only; after thatit is enough to have 2 firefighters per each round. This result significantly improves theknown solutions to a problem by Cai and Wang (there was no positive bound knownfor the surviving rate of general planar graph with only 3 firefighters). The proof isdone using the separator theorem for planar graphs. Introduction
The following
Firefighter Problem was introduced by Hartnell [6]. Consider any con-nected graph, say G, on n vertices, n ≥ . Let k be any positive integer. Suppose thata fire breaks out at some vertex v ∈ V ( G ) . Then in each turn firefighters can protect k vertices of G, not yet on fire and the protection is permanent. Next the fire spreads toall the unprotected vertices that are adjacent to some vertices already on fire. The goalis to save as much as possible and the question is how many vertices can be saved. Wewould like to refer the reader to the survey by Finbow and MacGillivray [3] for moreinformation on the background of the problem and directions of its consideration.In this paper we focus on the following aspect of the problem. Let sn k ( G, v ) denotethe maximum number of vertices of G that k firefighters can save when the fire breaksout at the vertex v. This parameter may depend heavily on the choice of the startingvertex v, for example when the graph G is a star. Therefore Cai and Wang [1] introducedthe following graph parameter: the k -surviving rate ρ k ( G ) is the expected fraction ofvertices that can be saved by k firefighters, provided that the starting vertex is chosenuniformly at random. Namely ρ k ( G ) = 1 | V ( G ) | X v ∈ V ( G ) sn k ( G, v ) . While discussing the surviving rate, let us mention the recent results by Prałat [10,11] which have provided a threshold for the average degree of general graphs, whichguarantees a positive surviving rate with a given number of firefighters. To be more
Mathematics Subject Classification.
Key words and phrases. firefighter problem, surviving rate, planar graph. precise, for k ∈ N + let us define τ k = (cid:26) for k = 1 k + 2 − k +2 for k ≥ . Then, there exists a constant c > , such that for any ǫ > , any n ∈ N + and any graph G on n vertices and at most ( τ k − ǫ ) n/ edges one has ρ k ( G ) > c · ǫ > . Moreover, thereexists a family of graphs with the average degree tending to τ k and the k -surviving ratetending to 0, which shows that the above result is the best possible.The k -surviving rate is investigated for many particular families of graphs — we focushere on planar graphs. Cai and Wang [1] asked about the minimum number of firefighters k such that ρ k ( G ) > c for some positive constant c and any planar graph G. It is easy tosee that ρ ( K ,n ) −−−→ n →∞ , hence at least 2 firefighters are necessary. It is shown that 2 isthe upper bound for triangle-free planar graphs [2] and planar graphs without 4-cycles [8].So far, for the general planar graphs the best known upper bound for the number offirefighters is 4: Kong, Wang and Zhu [7] have shown that ρ ( G ) > for any planar graph G. Esperet, van den Heuvel, Maffray and Sipma [2] have shown that using 4 firefightersin the first round only and just 3 in the subsequent rounds it is also possible to save apositive fraction of any planar graph G, namely ρ , ( G ) > . We use the notation of ρ k,l and sn k,l to describe the model with k firefighters in the first round and l firefightersin the subsequent rounds.In this paper, we improve the above bounds by the following theorem: Theorem 1.1.
Let G be any planar graph. Then: (i) ρ , ( G ) > . (ii) ρ ( G ) ≥ ρ , ( G ) > . In other words, we show that with 3 firefighters in the first round and just 2 in thesubsequent rounds we can save at least vertices of a planar graph, while with oneextra firefighter in the first round we can increase the saved fraction to . The proof
The proof is done using the lemma given by Lipton and Tarjan to prove the separatortheorem for planar graphs [9]. The key lemma in their proof, slightly reformulated touse in the firefighter problem, is quoted below.
Lemma 2.1.
Let G be any n -vertex plane triangulation and T be any spanning tree of G. Then there exists an edge e ∈ E ( G ) \ E ( T ) such that the only cycle C in T + e hasthe property that the number of vertices inside C as well as outside C is lower than n. A similar approach — using the above lemma to the firefighter problem on planargraphs, was first applied by Floderus, Lingas and Persson [5], with a slightly differ-ent notation of approximation algorithms. The authors of [5] have proved a theoremanalogous to Lemma 2.2. For some more details see Section 3.The proof of Theorem 1.1 is presented in two steps — first we show that ρ , ( G ) > for any planar graph G , then that ρ , ( G ) > . At first let us note that the survivingrate is monotone (non-increasing) with respect to the operation of adding edges to thegraph. Hence, it is enough to prove the bounds given by Theorem 1.1 only for planetriangulations. Moreover, in the first step, depending on the number of firefighters, we
LANAR GRAPH IS ON FIRE 3 save 3 or 4 vertices respectively, which is enough to obtain the desired bounds for anyplanar graph on not more than 17 vertices.Let G be any n -vertex plane triangulation, where n ≥ . Suppose that the fire breaksout at a vertex r. Consider a tree T obtained by the breadth-first-search algorithmstarting from the vertex r. By Lemma 2.1 there is the edge e and the cycle C ⊆ T + e such that | C ∪ in C | > n and | C ∪ out C | > n, where in C and out C denote the setsof vertices inside the cycle C and outside the cycle C respectively. Note that in thecycle C there are at most 2 vertices at any given distance from r. This holds becauseevery edge of C , except one, belongs to the breadth-first-search tree. The firefighters’strategy depends on the cycle C. When the vertex r does not belong to the cycle then thefirefighters protect the vertices of C in the order given by the distance from the vertex r and save all the vertices in either C ∪ in C or C ∪ out C. When the vertex r belongs tothe cycle C, the firefighters still can protect the vertices of the cycle except the vertex r, but it may be not enough, as the fire may spread through the neighbours of r inside aswell as outside the cycle C. Because either in C or out C contains not more than (cid:4) deg r − (cid:5) neighbours of r we get immediately: Lemma 2.2.
Let G be any n -vertex plane triangulation, where n ≥ . Suppose that thefire breaks out at some vertex r. Then using (cid:4) deg r − (cid:5) firefighters at the first step and2 at the subsequent steps one can save more than n/ − vertices. To calculate the surviving rate ρ , ( G ) let us now partition the vertex set of the graph G into 3 sets: X = { v ∈ V ( G ) : deg( v ) ∈ { , }} , Y = { v ∈ V ( G ) : deg( v ) ∈ { , , }} and Z = { v ∈ V ( G ) : deg( v ) ≥ } . Obviously we have | Z | = n − | X | − | Y | . Since for theplane triangulation we have n > n −
12 = X v ∈ V ( G ) deg v ≥ | X | + 5 | Y | + 8 | Z | , then one has | Y | > n − | X | . Using 4 firefighters at the first step one can save n − vertices if r ∈ X, more than n/ − vertices if r ∈ Y and at least vertices if r ∈ Z. A simple calculation shows nowthat ρ , ( G ) > ( n − | X | + ( n/ − | Y | + 4 | Z | n > , which finishes the proof for the case with 4 firefighters.Let us start our proof for 3 firefighters with a simple observation derived from Lemma 2.2. Observation 2.3.
Let G be any n -vertex plane triangulation, where n ≥ . Let r ∈ V ( G ) be a vertex of degree at most . Then sn , ( G, r ) > (cid:26) n − if deg( r ) ≤ n/ − if deg( r ) ∈ { , } Dealing with vertices of degree higher than 5 is a bit more complicated. Of course,the firefighters still can save at least 3 vertices, but frequently enough it is possible tosave more.
PRZEMYSŁAW GORDINOWICZ
Lemma 2.4.
Let G be any n -vertex plane triangulation, where n ≥ . Let r ∈ V ( G ) bea vertex of degree or . Then either sn , ( G, r ) > (cid:26) n/ − if deg( r ) = 6 n/ − if deg( r ) = 7 or the vertex r has at least adjacent neighbours, say u and v, such that sn , ( G, u ) > n/ and sn , ( G, v ) > n/ . Proof.
Let G be any n -vertex plane triangulation, where n ≥ . Let r ∈ V ( G ) be avertex of degree 6 or 7. Consider a tree T obtained by the breadth-first-search algorithmstarting from the vertex r. By Lemma 2.1 there is the edge e and the cycle C ⊆ T + e such that | C ∪ in C | > n and | C ∪ out C | > n. If the vertex r has no more than oneneighbour either inside the cycle C or outside the cycle, then sn , ( G, r ) ≥ n/ − . Without loss of generality we may assume then that the vertex r has exactly 2 neigh-bours inside the cycle C and at least 2 neighbours outside. When the vertex r has degree6 we may assume additionally, without loss of generality, that the number of verticesinside C is not lower than the number of vertices outside, that is | C ∪ in C | > n. Notethat the terms „inside” and „outside” the cycle depend on a particular drawing of thetriangulation.Let u and v be the neighbours of the vertex r inside the cycle. Then one of the twofollowing cases occur:Case 1. In the graph G there exists a path from u or v to a vertex on the cycle containingvertices in increasing distance from r. Case 2. There is no such path.
Solid edges are the edges of the span-ning tree, bold if they belong to thecycle. Dashed edges are the edgesforming the path (possibly but notnecessarily belonging the tree).
Figure 1.
Ilustration of Case 1.The path described in the first case divides the cycle C into two cycles C ′ and C ′′ (seeFigure 1), both of which have the properties that there are at most 2 vertices at anygiven distance from r and there is at most one neighbour of the vertex r inside the cycle.So, with 3 firefighters in the first round and 2 in the subsequent rounds one is able tosave every vertex except r from C ′ ∪ in C ′ or from C ′′ ∪ in C ′′ . Choosing the larger pieceit is possible to save at least half of the vertices from the set C ∪ in C (note that thevertices on the path count for both pieces). LANAR GRAPH IS ON FIRE 5
Considering the second case, note that as G is a triangulation then u and v areadjacent. Suppose now that the fire breaks out at the vertex u or v instead of thevertex r. The firefighters can now save all the vertices in C ∪ out C by protecting in thefirst round vertex r and its neighbours on the cycle C, and the vertices of C ordered inincreasing distance from r in the subsequent rounds. The fire cannot reach the verticeson the cycle C earlier than the firefighters protect them — otherwise, there would exist avertex on the cycle C which is closer to u or v than to r — this guarantees the existenceof a path described in Case 1. (cid:3) Let us now partition the vertex set of G into 4 subsets defined by the conditions: X = { v ∈ V ( G ) : sn , ( G, v ) > n − } ,Y = { v ∈ V ( G ) : deg( v ) ≤ ∧ sn , ( G, v ) ≤ n } ,Z = { v ∈ V ( G ) : deg( v ) ≥ ∧ sn , ( G, v ) ≤ n } ,W = V ( G ) \ ( X ∪ Y ∪ Z ) . We have that ρ , ( G ) ≥ n (cid:16) | X | ( n/ −
1) + 3 | Y | + 3 | Z | + | W | n/ (cid:17) . (1)Note that X contains every vertex of G with degree lower than 6. As the average degreeof vertex in a plane triangulation is lower than 6 then the set X is nonempty and theaverage degree of vertex in X is also lower than . Every vertex v ∈ W has deg( v ) ≥ and sn , ( G, v ) > n . By Lemma 2.4 every vertex in the set Y has at least 2 adjacentneighbours in the set X, hence we have | Y | ≤ P x ∈ X (deg x − P x ∈ X deg x − | X | . As any vertex in the set Z has degree at least 8, while the average degree in G is lowerthan 6, we have | Z | < P x ∈ X (6 − deg( x ))2 . These inequalities, when added, yield | Y | + | Z | < | X | . Hence, if only | Y | + 3 | Z | ≥ | X | , Inequality (1) yields ρ , ( G ) > | X | + | W | | X | + | W | = 221 . Note that it may occur that | Y | + 3 | Z | < | X | — this is an easier case as then ρ , ( G ) > n | X | − | X | + n | W | n ( | X | + | W | ) > | X | + | W | | X | + | W | = 221 . Remarks
While investigating whether or not the separator theorem had been used for thefirefighter problem the author found [4] (recently it was published in the journal versionas [5]). There a theorem analogous to Lemma 2.2 is proved. Moreover, [4, 5] present atheorem which in our notation would give that for any planar graph G and any vertex r one has sn ( G, r ) ≥ n r . Unfortunately, there is a serious error in the proof: it is
PRZEMYSŁAW GORDINOWICZ implicitly assumed that adding some edge joining two vertices of some induced subgraphof the planar graph, which preserves planarity of the subgraph, should also preserveplanarity of the whole graph (or, in other words, that the separator of the subgraph isalso a separator of the whole graph). In our opinion, such a result cannot be provedjust by a simple application of the separator theorem. Hence, the problem to determinewhether the -surviving rate of the planar graph may be bounded by some positiveconstant remains still open. References [1] L. Cai, W. Wang, The surviving rate of a graph for the firefighter problem,
SIAM J. Discrete Math. (2009) 1814-–1826.[2] L. Esperet, J. van den Heuvel, F. Maffray, F. Sipma, Fire containment in planar graphs J. GraphTheory (2013) 267–279.[3] S. Finbow, G. MacGillivray, The firefighter problem: a survey of results, directions and questions, Australasian Journal of Combinatorics (2009) 57—77.[4] P. Floderus, A. Lingas, M. Persson, Towards more efficient infection and fire fighting, Proceedings ofthe Seventeenth Computing: The Australasian Theory Symposium (CATS 2011), Perth, Australia,69–73.[5] P. Floderus, A. Lingas, M. Persson, Towards more efficient infection and fire fighting, InternationalJournal of Foundations of Computer Science (2013) 3-–14.[6] B. Hartnell, Firefighter! An application of domination, Presentation at the 25th Manitoba Confer-ence on Combinatorial Mathematics and Computing, University of Manitoba, Winnipeg, Canada,1995.[7] J. Kong, W. Wang, X. Zhu, The surviving rate of planar graphs, Theoret. Comput. Sci. , (2012)65-–70.[8] J. Kong, W. Wang, L. Zhang, The 2-surviving rate of planar graphs without 4-cycles, Theoret.Comput. Sci. (2012) 158–165.[9] R. J. Lipton, R. E. Tarjan, A Separator Theorem for Planar Graphs,
SIAM Journal on AppliedMathematics (1979), 177-–189.[10] P. Pralat, Graphs with average degree smaller than 30/11 burn slowly, Graphs and Combinatorics , (2014), 455–470.[11] P. Pralat, Sparse graphs are not flammable, SIAM Journal on Discrete Mathematics (2013),2157–2166. Institute of Mathematics, Technical University of Lodz, Łódź, Poland
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