A graph G is $k$-dot-critical (totaly $k$-dot-critical) if $G$ is dot-critical (totaly dot-critical) and the domination number is $k$. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a $k$-dot-critical graph and a totally $k$-dot-critical graph $G$ with no critical vertices for $k \geq 4$? We find the best bound for the diameter of a $k$-dot-critical graph, where $k \in\{4,5,6\}$ and we give a family of $k$-dot-critical graphs (with no critical vertices) with sharp diameter $2k-3$ for even $k \geq 4$

Doost Ali Mojdeh

Somayeh Mirzamani

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Opuscula Mathematica

On the diameter of dot-critical graphs

A graph G is k-dot-critical (totally k-dot-critical) if G is dot-critical (totally dot-critical) and the domination number is k. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306(2006), 11-18] the following question is posed: What are the best bounds for the diameter of a k-dot-critical graph and a totally k-dot-critical graph G with no critical vertices for k &ge; 4? We find the best bound for the diameter of a k-dot-critical graph, where k &isin; {4, 5, 6} and we give a family of k-dot-critical graphs (with no critical vertices) with sharp diameter 2k - 3 for even k &ge; 4

Mojdeh, D. A.

Mirzamani, S.

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Opuscula Mathematica  Vol. 29  No. 2  2009Doost Ali Mojdeh, Somayeh MirzamaniON THE DIAMETER OF DOT-CRITICAL GRAPHSAbstract. A graph G is k-dot-critical (totaly k-dot-critical) if G is dot-critical (totalydot-critical) and the domination number is k. In the paper [T. Burtona, D. P. Sumner,Domination dot-critical graphs, Discrete Math, 306(2006), 11{18] the following question isposed: What are the best bounds for the diameter of a k-dot-critical graph and a totallyk-dot-critical graph G with no critical vertices for k  4? We nd the best bound for thediameter of a k-dot-critical graph, where k 2 f4;5;6g and we give a family of k-dot-criticalgraphs (with no critical vertices) with sharp diameter 2k   3 for even k  4.Keywords: dot-critical graph, domination, diameter.Mathematics Subject Classication: 05C69.1. INTRODUCTIONLet G = (V;E) be a graph. A set S  V is a dominating set if every vertex of V nS isadjacent to some vertex of S. If S has the smallest possible cardinality of among alldominating sets of G, then S is called a minimum dominating set (abbreviated MDS)of G. The cardinality of any MDS of G is called the domination number of G and isdenoted by (G) [5] . More generally, we say that a set of vertices A dominates the setof vertices B if every vertex of B nA is adjacent to some vertex of A. Two graphs G1and G2 are disjoint if they have no vertex in common and no vertex of G1 is adjacent toany vertex of G2. We denote the open neighborhood of a vertex v in G by NG(v) andits closed neighborhood by NG[v] (so we have NG[v] = NG(v)[fvg). We indicate thefact that the vertex v is adjacent to a vertex u by writing v $ u. We denote the edgewith endpoints v and u by vu, the diameter of G by d = d(G) and the length of thepath with endpoints v and w with d(v;w). Let Ai and Aj be the sets of vertices. Weindicate the fact that every element of Ai is adjacent to every element of Aj by writingAi $ Aj and that the induced subgraph hAii is clique by writing Aic. If a property ofgraphs is worth studying, then it is almost certainly worthwhile to investigate thosegraphs that are extreme with respect to that property. But there may be many waysin which a graph can be extreme. In particular, for the domination number, there are165166 Doost Ali Mojdeh, Somayeh Mirzamania variety of extremal concepts that have been investigated. The two most studied arethe edge-critical graphs introduced by Sumner and Blitch [6] and the vertex-criticalgraphs introduced by Brigham et al [1] . A graph G is edge-critical with respect to thedomination number if for every two non-adjacent vertices v and u, (G+vu) < (G).A vertex v of G is critical if (G   v) < (G). A graph G is vertex-critical if everyvertex of G is critical. We denote the set of critical vertices of G by V  . A vertex v isstable if (G v) = (G). In [3,4] a new critical condition for the domination numberhas been introduced. A graph is domination dot-critical (hereafter, just dot-critical)if identifying any two adjacent vertices (i.e., contracting the edge comprising thosevertices) results in a graph with smaller domination number. If identifying any twovertices of G causes the domination number to decrease, then we say that G is totallydot-critical. When we say that G is k-edge-critical, k-vertex-critical, k-dot-critical, ortotally-k-dot-critical, we mean that it has the indicated property and that (G) = k.In the paper [4] T. Burton, and D.P. Sumner posed the question: What are the bestbounds for the diameter of a k-dot-critical graph and a totally k-dot-critical graphG with no critical vertices for k  4? We nd the best bound for the diameter of ak-dot-critical graph, where k 2 f4;5;6g, and we give a family of k-dot-critical graphs(with V   = ;) with sharp diameter 2k 3 for even k  4. We believe this bound canbe 2k   1 for odd k  5. The paper ends with some open problems.Note that, after we had sent the paper to Opuscula Mathematica for reviewingwe became aware of the papers ([\Domination dot-critical graphs with no criticalvertices", Zhao Chengye, Yang Yuansheng, Sun Linlin, Discrete Math. Volume 308,Issue 15 (2008)] and [\On the diameter of a domination dot-critical graph", NaderJafari Rad, Discrete Applied Mathematics 157 (2009), 1647-1649]). These papers,especially the second one, have some similar results, though we are sure that thesecond mentioned paper has been prepared after ours.The following facts are useful.Lemma A ([4]) If G is any graph with (G) = k  2, then G is dot-critical (resp.totally dot-critical) if and only if every two adjacent non-critical vertices (resp. anytwo non-critical vertices) belong to a common MDS.Lemma B ([4]) Let G be a dot-critical graph, v and u be two vertices of G. IfNG[v]  NG[u], then v 2 V  .2. EXAMPLESWe give some examples of dot-critical and totally dot-critical graphs:1. P3k+1 is dot-critical.2. C3k+1 is totally dot-critical.3. Let Kt be a complete graph and KtKt be the Cartesian product of Kt withitself (see Fig. 1). In [2] R.C. Brigham et al. have shown that the graph KtKtis t-vertex critical for t  3. Hence by Lemma A, KtKt, (t  3) is totallyt-dot-critical, because the graph does not have any non-critical vertices.On the diameter of dot-critical graphs 1674. The circulant graph C12h1;4i (see Fig. 2) is the graph with vertex setfv0;v1; ;v11g and edge set fvivi+j(mod 12)j i 2 f0;1; ;11g and j 2 f1;4gg.The graph G = C12h1;4i has domination number 4, and fv0;v3;v6;v9g is an MDSof G. The set of vertices fv3;v6;v9g dominates G fv0g and G is vertex transitive,therefore G is critical. By Lemma A, G is totally 4-dot-critical.Fig. 1. The graph K3K3Fig. 2. The circulant C12h1;4i5. The following gures (see Fig. 3) denote some examples of dot-critical graphs withno critical vertices. Each of them is constructed as follows. Let G = (V;E) be a168 Doost Ali Mojdeh, Somayeh Mirzamanigraph whereV = fu1;u2; ;un;u11;u12; ;u1k;u21;u22; ;u2k; ;un1;un2; ;unkgandE = fuiui 1;uiui+1;uiu(i 1)j;uiu(i+1)j;uilu(i 1)j;uilu(i+1)jj1  i(mod;n)  nand 1  j;l  k; where k  2g.For n 2 f4;5g and k = 2, see Fig. 3.Fig 3.3. MAIN RESULTSThe diameters of k-dot-critical graphs (k  4 ) with no critical vertices are studied.Theorem 3.1. A connected 4-dot-critical graph with V   = ;, has diameter of atmost ve.Proof. Let G be a connected 4-dot-critical graph with V   = ; and diameter d. Letv 2 V (G) be a vertex so that there is a vertex w 2 V and d(v;w) = d. Let Ai =fx 2 V (G)jd(v;x) = ig for 0  i  d. Then Ai 6= ; and A0 = fvg. For any x 2 A1,v $ x. The assumption says that any MDS has 4 vertices and according to Lemma Aany two adjacent vertices belong to an MDS. Let x 2 A1, and suppose there is anMDS containing the vertices v and x. Since these two vertices dominate at mostA0 [A1 [A2, so two other vertices of the MDS, s3 and s4 say, dominated Si=3Ai. Eachsi can dominate at most three of Ai's for 3  i  d. Hence it results that d  8. Thefollowing facts show that d 6= 6;7;8.Fig 4.Fact 1.1. Suppose d = 8. Let y 2 A2, z 2 A3 and y $ z. There exists anMDS of G containing both of y and z. Thus two other vertices of the MDS satisfyOn the diameter of dot-critical graphs 169fs3;s4g 8 Si=0i6=2;3Ai. Let s3 2 A0 [ A1 and suppose that8 Si=4Ai contains the vertex s4which dominates at most three of Ai's. This implies that s4 2 A6 and then A8 is notdominated by the MDS. Hence d  7.Fact 1.2. Suppose d = 7. If we consider the MDS of Fact 1.1, then for any w 2 A7,NG[w]  NG[s4] and so w 2 V   contradicts Lemma B. Thus d  6.Fact 1.3. Suppose d = 6. There are some cases.Case 1.1. Let MDS be the set S = fy;z;s3;s4g where y 2 A2 and z 2 A3, thenfs3;s4g 6 Si=0i6=2;3Ai. Suppose s3 2 A0 [ A1. Then the vertex s4 dominates A5 [ A6.If s4 2 A5, then for every w 2 A6, NG[w]  NG[s4], which implies w 2 V  , acontradiction. Hence s4 2 A6. We claim that A6 = fs4g. Let u 2 A6 n fs4g, thenNG[u]  NG[s4] and u 2 V  , a contradiction, so A6 = fs4g and A5 $ A6. Thusthe only element of S that can dominate A4 is z 2 A3 and since z has been chosenarbitrary, it means that A4 $ A3.Case 1.2. Let MDS be the set S = fz;u;s3;s4g where z 2 A3 and u 2 A4; thens3 2 A5 [ A6 and by Lemma B s4 = v 2 A0. Thus the only element of S that candominate A2 is z 2 A3 and since z has been chosen arbitrary, it means that A2 $ A3.Case 1.3. Let MDS be the set S = fv;x;s3;s4g where v 2 A0 and x 2 A1, thus6 Si=2Aicontains the vertices of the set fs3;s4g. There are some subcases.Subcase 1.3.1. If s3 2 A2 [ A3 and s4 2 A5, then for any w 2 A6, NG[w]  NG[s4]and so w 2 V   contradicts Lemma B.Subcase 1.3.2. Suppose that s3 2 A3; s4 2 A6, A6 has only one element by Case1.1 and A5 is dominated by s4. It is clear that any vertex z 2 A3 is dominated bys3 2 A3 and since s3 is arbitrary, then A3 is clique. This with together Cases 1.1 and1.2 implies that4 Si=2Ai is dominated by s3. Since the vertex v dominates A1, henceS = fv;s3;s4g is an MDS, a contradiction.Subcase 1.3.3. Suppose that s3 2 A4, s4 2 A5 [A6, so the only element of MDS thatcan dominate A2 is x 2 A1 and since x is an arbitrary, it means that A1 $ A2.Case 1.4. Let MDS be the set S = fu;w;s3;s4g where u 2 A5 and w 2 A6. Thereare some subcases.Subcase 1.4.1. Suppose s3 2 A4, so s4 must be in A1 and then NG[v]  NG[s4] andv 2 V  , a contradiction.Subcase 1.4.2. Suppose s3 2 A3 and s4 2 A0 [ A1, then A3 is clique and usingCases 1.1, 1.2 and 1.3 implies A2 $ Ac3 $ A4. Now fs4;s3;wg is an MDS of G, acontradiction.Subcase 1.4.3. Suppose s3 2 A2 and s4 2 A0 [ A1, then u $ A4. The vertex u 2 A5is arbitrary, so A4 $ A5.Using Cases 1.1{1.4, we have a contradiction or A0 $ A1 $ A2 $ A3 $ A4 $A5 $ A6. We show that the last relation leads to a contradiction. For this, let170 Doost Ali Mojdeh, Somayeh MirzamaniS = fy;z;wg where y 2 A2, z 2 A3 and w 2 A6, then fy;zg dominates4 Si=1Ai and wdominates A5 [ A6. Hence for v 2 A0 the set S = fy;z;wg is an MDS for G   v andv 2 V  , a contradiction. These contradictions show that d  5.The bound on a diameter in Theorem 1 is sharp, see the following example.Example 1. The graph G in Figure 5 is a 4-dot-critical with V   = ; and diameterd = 5.Fig 5.Theorem 3.2. A connected 5-dot-critical graph with V   = ; has a diameter of atmost seven.Proof. Let G be a connected 5-dot-critical graph with V   = ; and diameter d. Letv and w 2 V (G) be such that d(v;w) = d. By Theorem 1, Ai 6= ; and A0 = fvg.For any x 2 A1, v $ x and there is an MDS of G containing both v and x soalso fv;x;s3;s4;s5g. Thus,d Si=3Ad is dominated by three vertices fs3;s4;s5g. Henced  11. Through the following facts we show that d 6= 8;9;10;11.Fact 2.1. Suppose d = 11. For u 2 A4 and w 2 A5 there is an MDS of G containingboth u and w. If A0 [ A1 [ A2 is dominated by one vertex of this MDS, s3 say,then s3 2 A1, and NG[v]  NG[s3] and so v 2 V  , a contradiction. Hence3 Si=0Ai isdominated by two vertices fs3;s4g. Thusd Si=7Ai is dominated by one vertex s5. Henced  9.Fact 2.2. Suppose that d = 9 and S = fu;w;s3;s4;s5g is an MDS where u 2 A4 andw 2 A5. In the same way as in Fact 2.1, fs3;s4g dominates3 Si=0Ai and s5 2 A8. Thenfor any u 2 A9, NG[u]  NG[s5] and u 2 V  , a contradiction. Hence d  8.Fact 2.3. Suppose that d = 8 and MDS is the set S = fu;w;s3;s4;s5g. There aresome cases.Case 2.1. Let u 2 A4 and w 2 A5, then fs3;s4g 3 Si=0Ai. If s5 2 A7, then for anyu 2 A8, NG[u]  NG[s5] and u 2 V  , a contradiction. If s5 2 A8 and there is a vertexOn the diameter of dot-critical graphs 171(s5 6=)y 2 A8, then NG[y]  NG[s5], so y 2 V  , a contradiction. Hence A8 = fcg,A7 $ A8 and A5 $ A6 because w 2 A5 is an arbitrary vertex that dominates A6.Case 2.2. Let u 2 A3, w 2 A4, then fs3;s4g 8 Si=5Ai and s5 = v 2 A0. So udominates A2 and A2 $ A3, because u 2 A3 is an arbitrary vertex that dominatesA2.Case 2.3. Let u 2 A2 and w 2 A3, then s3 2 A0 [ A1 and8 Si=5Ai is dominated by twovertices.Case 2.4. Let u 2 A5 and w 2 A6, then s5 2 A7 [ A8 and3 Si=0Ai is dominated by twovertices.Case 2.5. Let u(= v) 2 A0 and w 2 A1, then fs3;s4;s5g 8 Si=2Ai. The followingsubcases may take place.Subcase 2.5.1. Let s3 2 A2, s4 2 A5 and s5 2 A8, then8 Si=4Ai is dominated by twovertices. Now using Case 2:4 one can conclude that8 Si=0Ai is dominated by 4 vertices,a contradiction.Subcase 2.5.2. Let s3 2 A3, s4 2 A5 and s5 = c 2 A8. These assumptions and Case2.2 imply that the set fs3;s4;s5g dominates8 Si=2Ai. Therefore S = fu;s3;s4;s5g isan MDS of G, a contradiction.Subcase 2.5.3. Let s3 2 A3; s4 2 A6 and s5 = c 2 A8. Then similarly as in Subcase2.5.2, S = fu;s3;s4;s5g is an MDS of G, a contradiction.Subcase 2.5.4. Let fs3;s4;s5g 8 Si=4Ai, then w dominates A2. Since w is an arbitraryvertex of A1, hence A1 $ A2. Now we immediately ash back to Case 2.3 wherethe MDS is S = fu0;w0;s03;s04;s05g with u0 2 A2, w0 2 A3 and s03 2 A0, and onecan see that the set S = fu0;w0;s04;s05g dominates8 Si=1Ai. Hence (G   v) = 4 forv 2 A0 = fvg, a contradiction. Since there is an MDS with two adjacent verticesu = v 2 A0 and w 2 A1 and when d = 8 this anyway leads to a contradiction. Thusd  7.Problem 1. Is there a connected 5-dot-critical graph with V   = ; and d = 7?Theorem 3.3. A connected 6-dot-critical graph with V   = ; has a diameter of atmost nine.Proof. Let G be a connected 6-dot-critical graph with V   = ; and diameter d. Letv;w 2 V (G) be such that d(v;w) = d. Let Ai = fx 2 V (G)jd(v;x) = ig for 0  i  d.So A0 = fvg and for x 2 A1, v $ x. By Lemma A, there exists an MDS of Gcontaining both v and x, like fv;x;s3;s4;s5;s6g. Thusd Si=3Ai are dominated by four172 Doost Ali Mojdeh, Somayeh Mirzamanivertices fs3;s4;s5;s6g. Hence d  14. Through the following facts we show thatd 6= 10;11;12;13;14.Fact 3.1. Suppose that d = 14, u $ w and S = fu;w;s3;s4;s5;s6g is an MDS ofG. Let u 2 A4 and w 2 A5. Then A0 [ A1 [ A2 is not dominated by one vertexs3 2 A1, because we will have NG[v]  NG[s3] for v 2 A0 and then v 2 V  , acontradiction. Hence fs3;s4g 4 Si=0Ai andd Si=7Ai is dominated by two vertices offs5;s6g, a contradiction. Hence d  12.Fact 3.2. Suppose that d = 12 and S = fu;w;s3;s4;s5;s6g is an MDS of G whereu 2 A4 and w 2 A5. Then fs3;s4g 4 Si=0Ai, s5 2 A8 and s6 2 A11. If x 2 A12, thenNG[x]  NG[s6] and so x 2 V  , a contradiction, Hence d  11.Fact 3.3. Suppose that d = 11 and fu;w;s3;s4;s5;s6g is an MDS of G. There aresome cases.Case 3.1. Let u 2 A4 and w 2 A5; then fs3;s4g 3 Si=0Ai, s5 2 A8 and s6 2 A11. Ifthere is a vertex y 2 A11 s6, then NG[y]  NG[s6], and so y 2 V  . Thus A11 = fcg.Therefore we have A5 $ A6, A11 $ A10 and11 Si=7Ai is dominated by two vertices.Case 3.2. Let u 2 A6 and w 2 A7 then fs3;s4g 11 Si=8Ai and s5 2 A0 and s6 2 A3.Then4 Si=0Ai is dominated by two vertices.Case 3.3. Let u 2 A2, w 2 A3 then s3 2 A0 [ A1, so we consider the followingsubcases.Subcase 3.3.1. If s4 2 A5; s5 2 A7, then s6 2 A10 and c 2 A11 would be a criticalvertex, a contradiction.Subcase 3.3.2. s4 2 A5; s5 2 A8 and s6 2 A11 = fcg.Subcase 3.3.3. s4 2 A6; s5 2 A8 and s6 2 A11 = fcg.Subcase 3.3.4. s4 2 A6; s5 2 A9 and s6 2 A10.Subcase 3.3.5. s4 2 A6; s5 2 A9 and s6 2 A11 = fcg.These Subcases result in a contradiction or A5[A6 is dominated by one vertex s4.Now combining Cases 3:1, 3:2 and 3:3 implies that the graph G is dominated by 5vertices, a contradiction. Hence d  10.Fact 3.4. Suppose that d = 10 and fu;w;s3;s4;s5;s6g is an MDS of G. There aresome cases.Case 3.4. Let w 2 A7 and u 2 A8, then s3 2 A9 [ A10 and5 Si=0Ai is dominated bythe set fs4;s5;s6g.Case 3.5. Let w 2 A3 and u 2 A4, then we have the following subcases.Subcase 3.5.1. If A0[A1[A2 contains 2 vertices of the MDS, then10 Si=6Ai is dominatedby 2 vertices. This result combined with Case 3.4 yields that G has an MDS withsize 5, a contradiction.On the diameter of dot-critical graphs 173Subcase 3.5.2. Suppose that A0 [ A1 [ A2 contains one vertex s3 of the MDS.If s3 2 A1, then NG[v]  NG[s3] and so v 2 V  , a contradiction.If s3 = v 2 A0, then w $ A2. Since w is an arbitrary vertex in A3, then A2 $ A3Case 3.6. Let u 2 A5 and w 2 A6, then3 Si=0Ai is dominated by 2 vertices and so does10 Si=8Ai by 2 other vertices of the MDS.Case 3.7. Let u = v 2 A0 and w 2 A1, then A2 [ A3 [ A4 contains one vertexs3 of MDS (otherwise10 Si=6Ai is dominated by 2 vertices and by Case 3.4 we have acontradiction). We consider the following subcases.Subcase 3.7.1. Suppose that s3 2 A2, so10 Si=4Ai is dominated by 3 vertices. This resultcombined with Case 3.6 yields that G has an MDS with size 5, a contradiction.Subcase 3.7.2. Suppose that s3 2 A3. Since A2 $ A3 and A0 $ A1, then one canreplace fu;w;s3g with fu;s3g. Hence G has an MDS with size 5, a contradiction.Subcase 3.7.3. Suppose that s3 2 A4. Then s3 $ A3 and w $ A2. Since w is anarbitrary vertex, then A1 $ A2.Case 3.8. Let u 2 A2 and w 2 A3, then u dominates A1 (Subcase 3.7.3) and one canchoose s3 = v 2 A0. Thus (G   v) = 5 and v 2 V  , a contradiction. These Casesshow that there a 6-dot critical graph with diameter 10 does not exist. Therefored  9 and the proof is completed.The bound on diameter in Theorem 3 is sharp, see the following example.Example 2. The graph G in Figure 6 is a 6-dot-critical with no critical vertices anddiameter d = 9.Fig 6.We will show a family of dot-critical graphs with sharp diameter, see below.Proposition 3.4. Let n  4 be an even number. There is a family of n-dot-criticalgraph G with V   = ; and diameter d = 2n   3.Proof. For n = 4 and n = 6 Figures 5 and 6 show the necessary result. Let G bea graph with vertex set fvi1;vi2; ;vi(4m 2)j 1  i  k and k  3g and edge setfvijvl(j+1)j 1  i;l  k and 1  j  4m   3g (see Fig. 7) where m  4. If n = 2m,then the set S = f4t + 2;4t + 3j 0  t  m   2g [ f4m   3;4m   2g is an MDS ofG. It is easily seen that G is dot-critical, V   = ; and its diameter is 2m   3. The174 Doost Ali Mojdeh, Somayeh Mirzamanidierent values of k give us a family of n = 2m-dot-critical graphs with V   = ; anddiameter d = 4m   3.Example 3. The graph below (Fig. 7) is a connected 8-dot-critical graph withV   = ; and diameter d = 13.Fig 7.Problem 2. Is there a connected n = 2m + 1-dot-critical graph with V   = ; anddiameter d = 2n   3?The studied results make us believe that for any n  7, using the method of theproofs of Theorems 1-3 with more cases and Proposition 3.4 will give us the similarresults.Our belief is posed as a conjecture.Conjecture. For n  9, a connected n-dot-critical graph with V   = ; has a diameterof at most 2n   3.AcknowledgementThe authors would like to thank the referee for many helpful suggestions.The research was in part supported by a grant from IPM (No. 85050036).REFERENCES[1] R.C. Brigham, P.Z. Chinn, R.D. Dutton, A study of vertex domination critical graphs,Technical Report, University of Central Florida, 1984.[2] R.C. Brigham, T.W. Hanaynes, M.A. Henning, D.F. Rall, Bicritical domination, DiscreteMathematics 305 (2005), 18{32.[3] T.A. Burton, Domination dot-critical graphs, Ph.D. Dissertation, University of SouthCarolina, 2001.[4] T.A. Burton, D.P. Sumner, Domination dot-critical graphs, Discrete Mathematics 306(2006), 11{18.[5] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Fundamentals of Domination inGraphs, Marcel Dekker, Inc., NewYork, 1998.[6] D.P. Sumner, P. Blitch, Domination critical graphs, JCT Ser. B 34 (1983), 65{76.On the diameter of dot-critical graphs 175Doost Ali Mojdehdamojdeh@yahoo.comInstitute for Studies in Theoretical Physics and Mathematics (IPM)Tehran, IRIDepartment of MathematicsUniversity of MazandaranBabolsar, IRI, P.O. Box 47416-1467Somayeh MirzamaniDepartment of MathematicsUniversity of MazandaranBabolsar, IRI, P.O. Box 47416-1467Received: September 2, 2007.Revised: February 2, 2009.Accepted: April 22, 2009.

On the diameter of dot-critical graphs

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On the diameter of dot-critical graphs

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