Isometric-path numbers of block graphs

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Information Processing Letters 93 (2005) 99–102

www.elsevier.com/locate/ipl

Isometric-path numbers of block graphs

✩

Jun-Jie Pan

a

_{, Gerard J. Chang}

b,c,∗

a_{Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan} b_{Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan}

c_{Mathematics Division, National Center for Theoretical Sciences at Taipei, Old Mathematics Building,}

National Taiwan University, Taipei 10617, Taiwan Received 4 November 2003

Communicated by A.A. Bertossi

Abstract

An isometric path between two vertices in a graph G is a shortest path joining them. The isometric-path number of G, denoted by ip(G), is the minimum number of isometric paths required to cover all vertices of G. In this paper, we determine exact values of isometric-path numbers of block graphs. We also give a linear-time algorithm for finding the corresponding paths.

Keywords: Isometric path; Block graph; Cut-vertex; Algorithms

1. Introduction

An isometric path between two vertices in a graph

G is a shortest path joining them. The isometric-path

number of G, denoted by ip(G), is the minimum

num-ber of isometric paths required to cover all vertices of G. This concept has a close relationship with the game of cops and robbers described as follows. The game is played by two players, the cop and the

rob-ber, on a graph. The two players move alternatively,

✩

Supported in part by the National Science Council under grant NSC90-2115-M002-024.

* _{Corresponding author.}

E-mail address: gjchang@math.ntu.edu.tw (G.J. Chang).

starting with the cop. Each player’s first move con-sists of choosing a vertex at which to start. At each subsequent move, a player may choose either to stay at the same vertex or to move to an adjacent ver-tex. The object for the cop is to catch the robber, and for the robber is to prevent this from happening. Nowakowski and Winkler [7] and Quilliot [8] inde-pendently proved that the cop wins if and only if the graph can be reduced to a single vertex by succes-sively removing pitfalls, where a pitfall is a vertex whose closed neighborhood is a subset of the closed neighborhood of another vertex. As not all graphs are cop-win graphs, Aigner and Fromme [1] introduced the concept of the cop-number of a general graph G, denoted by c(G), which is the minimum number of

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cops needed to put into the graph in order to catch the robber. On the way to giving an upper bound for the cop-numbers of planar graphs, they showed that a single cop moving on an isometric path P guar-antee that after a finite number of moves the robber will be immediately caught if he moves onto P . Ob-serving this fact, Fitzpatrick [4] then introduced the concept of isometric-path cover and pointed out that

c(G) ip(G).

The isometric-path number of the Cartesian prod-uct Pn1× Pn2 × · · · × Pnd has been studied in the

lit-erature. Fitzpatrick [5] gave bounds for the case when

n1= n2= · · · = nd. Fisher and Fitzpatrick [3] gave

exact values for the case d= 2. Fitzpatrick et al. [6] gave a lower bound, which is in fact the exact value if d+ 1 is a power of 2, for the case when n1= n2=

· · · = nd= 2.

The purpose of this paper is to give exact values of isometric-path numbers of block graphs. We also give a linear-time algorithm to find the correspond-ing paths. For technical reasons, we consider a slightly more general problem as follows. Suppose every ver-tex v in the graph G is associated with a non-negative integer f (v). We call such function f a vertex

label-ing of G. An f -isometric-path cover of G is a family

C of isometric paths such that the following conditions

hold.

(C1) If f (v)= 0, then v is in an isometric path in C. (C2) If f (v) 1, then v is an end vertex of at least

f (v) isometric paths inC, while the counting is

twice if v itself is a path inC.

The f -isometric-path number of G, denoted by

ip_f(G), is the minimum cardinality of an f

-isometric-path cover of G. It is clear that when f (v)= 0 for all vertices v in G, we have ip(G)= ipf(G). The attempt

of is paper is to determine the f -isometric-path num-ber of a block graph. Recall that a block graph is a graph in which every block is a complete graph. A

cut-vertex of a graph is a cut-vertex whose removal results in a

graph with more components than the original graph. It is well known that in a block graph all internal ver-tices of an isometric path are cut-verver-tices.

2. Block graphs

In this section, we determine the f -isometric-path numbers for block graphs G. Without loss of general-ity, we may assume that G is connected. First, a useful lemma.

Lemma 1. Suppose x is a non-cut-vertex of a block

graph G with a vertex labeling f . If vertex labeling

fis the same as f except that f(x)= max{1, f (x)},

then ip_f(G)= ipf(G).

Proof. As any internal vertex of an isometric path in

a block graph is a cut-vertex but x not a cut-vertex,

x must be an end vertex of any isometric path. It

fol-lows that a collectionC is an f -isometric-path cover if and only if it is an f-isometric-path cover. The lemma then follows. 2

So, now we may assume that f (v) 1 for all non-cut-vertices v of G, and call such a vertex labeling

regular. Now, we have the following theorem for the

inductive step.

Theorem 2. Suppose G is a block graph with a regular

labeling f , and x is a non-cut-vertex in a block B with exactly one cut-vertex y or with no cut-vertex in which case let y be any vertex of B− {x}. When f (x) = 1, let

G= G − x with a regular vertex labeling fwhich is the same as f except f(y)= f (y) + 1. When f (x)

2, let G= G with a regular vertex labeling fwhich is the same as f except f(x)= f (x) − 1 and f(y)=

f (y)+ 1. Then ipf(G)= ipf(G).

Proof. We first prove that ip_f(G) ipf(G).

Sup-pose C is an optimal f -isometric-path cover of G.

Choose a path P inC having x as an end vertex. We consider four cases.

Case 1.1. P = x and f (x) = 1 (i.e., G= G − x).

In this case, C= (C − {P }) ∪ {y} is an f -isomet-ric-path cover of G. Hence, ip_f(G)= |C| |C|

ip_f(G).

Case 1.2. P = x and f (x) 2 (i.e., G= G). In

this case, C= (C − {P }) ∪ {xy} is an f -isometric-path cover of G. Hence, ip_f(G) = |C| |C|

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Case 1.3. P= xz for some vertex z in B −{x, y}. In

this case,C= (C −{P })∪{xy} is an f-isometric-path cover of G. Hence, ip_f(G)= |C| |C| ipf(G).

Case 1.4. P= xyQ, where Q contains no vertices

in B. In this case,C= (C − {P }) ∪ {yQ} is an f -iso-metric-path cover of G. Hence, ip_f(G)= |C| |C|

ip_f(G).

Next, we prove that ip_f(G) ipf(G). SupposeC

is an optimal f-isometric-path cover of G. Choose a path PinC having y as an end vertex. We consider three cases.

Case 2.1. P= yx. In this case, G= G and C =

(C− {P}) ∪ {x} is an f -isometric-path cover of G.

Hence, ip_f(G) |C| |C| = ipf(G).

Case 2.2. P= yz for some z in B − {x, y}. In this

case, C = (C− {P}) ∪ {xz} is an f -isometric-path cover of G. Hence, ip_f(G) |C| |C| = ipf(G).

Case 2.3. P= yQ, where Q contains no vertex in

B. In this case,C = (C− {P}) ∪ {xyQ} is an f

-iso-metric-path cover of G. Hence, ip_f(G) |C| |C| =

ip_f(G).

Consequently, we have the following result for

f -isometric-path numbers of connected block graphs.

Theorem 3. If G is a connected block graph with a

regular vertex labeling f , then ip_f(G)= s(G)/2,

where s(G)=_v_{∈V (G)}f (v).

Proof. The theorem is obvious when G has only one

vertex. For the case when G has more than one ver-tex, we apply Theorem 2 repeatedly until the graph becomes trivial. Notice that s(G)= s(G) when

The-orem 2 is applied. 2

For the isometric-path-cover problem, we have

Corollary 4. If G is a connected block graph, then

ip(G)= nc(G)/2, where nc(G) is the number of

non-cut-vertices of G.

Proof. The corollary follows from Theorem 3 and the

fact that ip(G)= ipf(G) for the regular vertex

la-beling f with f (v)= 1 if v is a non-cut-vertex and

f (n)= 0 otherwise. 2

3. Algorithm

Based on Theorem 2, we are able to design an al-gorithm for the isometric-path-cover problem in block graphs. Notice that we may only consider connected block graphs with regular vertex labelings. To speed up the algorithm, we may modify Theorem 2 a little bit so that each time a non-cut-vertex is handled.

Theorem 5. Suppose G is a block graph with a regular

labeling f , and x is a non-cut-vertex in a block B with exactly one cut-vertex y or with no cut-vertex in which let y be any vertex in B− {x}. Let G= G − x with a regular vertex labeling f which is the same as f except f(y)= f (y)+f (x). Then ipf(G)= ipf(G).

Proof. The theorem follows from repeatedly applying

Theorem 2. 2

Now, we are ready to give the algorithm.

Algorithm PG. Find the f -isometric-path number ip_f(G) of a connected block graph.

Input. A connected block graph G and a regular vertex la-beling f .

Output. An optimal f -isometric-path cover C of G and ip_f(G).

Method.

1. construct a stack S which is empty at the beginning; 2. let G← G;

3. while (Ghas more than one vertex) do

4. choose a block B with exactly one cut-vertex y or with no cut-vertex in which case choose any y∈ B; 5. for (all vertices x in B− {y}) do

6. f (y)← f (y) + f (x); 7. push (x, y, f (x)) into S; 8. G← G− x;

9. end for; 10. end while;

11. ip_f(G)← f (r)/2, where r is the only vertex of G; 12. letC be the family of isometric paths containing ip(G)

copies of the path r ; 13. while (S is not empty) do 14. pop (x, y, i) from S;

15. choose i copies of path P inC using y as an end vertex;

16. if (P= yx) then

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18. if (P= yz for some vertex z in the block of G con-taining x) then

19. replace the i copies of P by i copies of xz inC; 20. if (P= yQ where Q has no vertices in the block of

G containing x) then

21. replace the i copies of P by the i copies of xyQ inC;

22. end while.

Algorithm PG can be implemented in time linear to the number of vertices and edges. Notice that we can use the depth-first search to find all blocks and cut-vertices of a graph in linear time, see [2].

Acknowledgements

The authors thank the referee for many constructive suggestions.

References

[1] M. Aigner, M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1984) 1–12.

[2] T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algo-rithms, MIT Press, Cambridge, MA, 1990.

[3] D.C. Fisher, S.L. Fitzpatrick, The isometric path number of a graph, J. Combin. Math. Combin. Comput. 38 (2001) 97–110. [4] S.L. Fitzpatrick, Aspects of domination and dynamic

domina-tion, Ph.D. Thesis, Dalhousie University, Nova Scotia, Canada, 1997.

[5] S.L. Fitzpatrick, The isometric path number of the Cartesian product of paths, Congr. Numer. 137 (1999) 109–119. [6] S.L. Fitzpatrick, R.J. Nowakowski, D. Holton, I. Caines,

Cov-ering hypercubes by isometric paths, Discrete Math. 240 (2001) 253–260.

[7] R. Nowakowski, P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983) 235–239.