图的连通性是图论的一个基本研究课题，它与网络的可靠性（容错性）密切相关.图的经典的连通性是用连通度和 边连通度来度量的.图的连通度（边连通度）就是使图不连通所要删除的最小的点（边）数，显然它反映了相应的网络的容错性. 在通信网络的可靠性的进一步研究中，人们需要了解两个具有相同的连通度或边连通度的图，哪个可靠性更高？ 为了深入研究网络的可靠性或容错性，人们提出了许多条件连通性的概念. Harary在1983年引入条件连通度的概念\cite{2}.设图$G=(V，E)$，$P$是图的某种性质，$S\subseteqV(G)$.$G$的条件连通度 $\kappa(G:P)$=min...The connectivity of graph is an element topic in the research of graph theory, it is closely related to the network reliability (fault tolerance). The classical connectivity concept of graph is connectivity and edge connectivity. Connectivity (edge connectivity) is the minimum number of vertices (edges) whose deletion makes the graph disconnected. Obviously, they reflect the corresponding net...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1912008015045

郭利涛

Xiamen University Institutional Repository

dH8~10384 ℄K) gd )19120080150451 UDCK ! e  y ?87j;CThe Conditional Connectivity of GraphsN p -l 1 d  O F E d$s R  ' X Y ) N{=/b 2011  `{=. 2011  `N;$℄ 2011  `=/Nz$ -; 3 f2011 , 3 1厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库Doctoral DissertationThe Conditional Connectivity of GraphsByLitao GuoSupervisor: Professor Xiaofeng GuoSpeciality: Graph TheoryInstitution: School of Mathematical SciencesXiamen UniversityXiamen, P. R. ChinaMarch, 2011厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库{y zx~|2ek~fz&2e8_+;W|jk"j2e8zLpY3?OweTab~KAk"j+8p (V`?Gd1MvS--	fk"f5Q7S+Vmykfz <_k"jR <_\Tp)|8 p)|jR8 |D*&fL<_T<_i=eTp)fÆ|&m8 OZDeD, 1 k厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库{y z}~|2em-	f}'p>e-fhU:V%M3*p-|fzGCyT	T?_3ZH~fz(Dbl#-,#4_fzr-	fs2d?9'/[222em-	f|fzor\L""fzz9'VtK|fz?_-CuO9s#WJTK?QV1Ogifz2fz3# 1. -	f*.M[+3*fz#  ,  1 k8(|7.Z  2. O*(|7.ZR8 |;D*& ” ”Tf|;&m*fz&~-	f*.M3"fzÆ-	f*.M3fz+/fz|HOfLi/fz+(|7.ZeD, 1 k厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库r <  P ixkt\kX't{|xX3l#=`R'}95XnXÆM<t+\kX'\k=.7\k=G=at\k=7\k=Æ%'tP\kMvzvG*7Æ:2bRQ<}nX9kP}95X|Rl#qf\v_x)<n\k=U7\k=t!x95X ssl#}95XUnXf℄t`Fix\kXn/Harary 9 1983 -six\k=n/ [38]. t G = (V, E)  P 'tuXm S ⊆ V (G). G ix\k= κ(G : P )=min{|S| : G − S P\kHO G − S x^V9Xm P}. ns94ix7\k=λ(G : P ). p o P x\k^Ve k x*Xm# λ(G : P )%'^> k- 7jX7\k= λk(G). LF%C$*QC94 k- 7jX\k= κk(G).  S ⊆ E(G) (V (G)), G − S P\kHO G − S x^Ve k x*g S  k- 7jX7 (*) uξk(G) = min{ |[U, U ]| : ∅ 6= U ⊂ V (G), |U | = k O G[U ] \k }. |xt G λk-p λk(G) = ξk(G). t G'b -λk (b -κk)\kp xG k- 7jX7u (*u) 9X|x k x*\ktt G 'b -κk\kp κk(G) = ξk(G) OxG k- 7jX*u9X|x k x*\kt (k = 1, 2, 3). [E% k = 1 gt G 'b7\kb\kÆ[E% P kδ Bt GW.|x7u F 9x\k^VG=δ 	$ $ k Xmix7\k= λkδ(G) '|x)Xm P kδ G7u F X: (p Mx7b9).Fábrega . Fiol s7 extra- \k= (edge extraconnectivity) n/[27].  P ke B\k^VF$ k x*Xmt G |x7b F '7extra- up G − F P\kO G − F x\k^V)Xm P ke . 7extra- \k= λke(G) 'G7 extra- uX: D 'K\kDt D ;b S '|x7jX;up D − S|xZ:OK\k^V D1  D − V (D1) )&|i;7jX;\k= λ′(D) 'G7jX;uX:3zvl#t 3-7jX7*Æ\k=X$tzu[4Z:厦门大学博硕士论文摘要库x  /MoM "M:z<t' λ3- b -λ3 \k.b -κ3 \kq^ixCDt7jX;\k=zvj*ixDt' λ′-q^ixFl#iu[1tix\kXeUX*='\kX'zv	 p,|t 3- 7jX7*Æ\k=k#l#[ `=be_x*	j*. 3- 7jX7\k='t' λ3-q^ix_x*	j*=be=. ixP&x Tt'b 3- 7jX7\kq^ixR=ixRt'b 3- 7jX7\kq^ixte9t[ .`'t'b -λ3 \k.b -κ3 \kq^ixJDt7jX;\k= D'K\kDtK\kDt D0g λ′-\kp D97jX;u xy' D|i;O yx ∈ A(D),g ξ′(xy) = min{d+(x)+d+(y)−1, d−(x)+d−(y)−1, d+(y)+d−(x), d+(x)+d−(y)−1}' xy ;=p xy ' D |i;O yx ∈ A(D), g ξ′(xy) = min{d+(x) + d+(y) − 2, d−(x) +d−(y)− 2, d+(y) + d−(x) − 1, d+(x) + d−(y) − 1} ' xy ;= D G;=' ξ′(D) = min{ξ′(xy) : xy ∈ A(D)}. |x λ′- \kDt0g λ′- p λ′(D) = ξ′(D).3_x*	j*ixDt' λ′-q^ixx[1tix\kX G = (V, E), t G  Mycielskian <H µ(G) *b V ∪ V ′ ∪ {u}, q V ′ = {x′ : x ∈ V } 7b E ∪ {xy′ : xy ∈ E} ∪ {y′u : y′ ∈ V ′}. * x′ ' x twin ( x y' x′  twin), O u0g µ(G)~C$ n ≥ 2, µn(G)4 µn(G) = µ(µn−1(G)). 4X µ(G)\kn x*℄'t Tn '}℄t Kn x*p}|xDMt4 Dn[G] = G ⊗ Tn.3l#t Mycielskian <H µ(G) b\kX µ(G) 'b\k.b7\kqvix Dn[G] 'b\kqvixl#JUt.}℄t Kronecker \b\kX}℄tYt.}℄tKronecker \b\kX^gtYt.}℄t Kronecker\b\kXTMJt'b\kl#bXW.I0bXWix\kXH|Jix\k=℄厦门大学博硕士论文摘要库r <  P xiD*='\kX'4X*t G *=R(G) =∑v∈V1d(v).3k#l#*=' n, δ, ξ 't'b7\k. λ′- q^ixSh69l=E k- Vb (#) =E# extra- =E厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库W <  P xiiiAbstractThe connectivity of graph is an element topic in the research of graph theory,it is closely related to the network reliability (fault tolerance). The classical connec-tivity concept of graph is connectivity and edge connectivity. Connectivity (edgeconnectivity) is the minimum number of vertices (edges) whose deletion makes thegraph disconnected. Obviously, they reflect the corresponding network fault toler-ance. For the further study of the network reliability, peoples need to know if twographs have the same connectivity or edge connectivity, which has the more relia-bility? For further study, many concepts of connectedness have been introduced.Harary proposed the conditional connectivity in 1983[38]. Let G be a graphand P be graph-theoretic property, S ⊆ V (G). The conditional connectivity of Gis κ(G : P )=min{|S| : G − S is not connected and each component of G − S hasproperty P}. Similarly we can define the conditional edge connectivity λ(G : P ).If let P be the property each component has at least k vertices, then λ(G : P ) isk-restricted edge connectivity λk(G). Similarly, for the case of vertex, we can definek-restricted connectivity κk(G). Graph G is λk-optimal, if λk(G) = ξk(G). Graph Gis super-λk (super-κk), if each minimum k-restricted edge (vertex) cut isolates oneconnected subgraph with k vertices. Particularly  if k = 1  we call G is superedge connected (super connected).In particular, let P kδ be the property of each connected component having min-imum degree δ ≥ k. Conditional edge connectivity λkδ(G) is the cardinality of theminimum edge cuts, if any, whose deletion disconnects G and each component ofG − F has property P kδ .Fábrega and Fiol introduced edge extraconnectivity in [27]. Let P ke be propertyof each connected component having more than k vertices. F ⊆ E is an edge set, Fis called edge extra-cut, if G−F is not connected and each component of G−F hasproperty P ke . The edge extraconnectivity λke(G) is the cardinality of the minimumedge extra-cuts.Let D be a strongly connected digraph. An arc set S of D is a restricted arc cutof D if D−S has a non-trivial strong component D1 such that D−V (D1) contains厦门大学博硕士论文摘要库xiv  /MoM "M:z<an arc. The restricted arc connectivity λ′(D) is the minimum cardinality over allrestricted arc cuts S.In this paper, we study the 3-restricted edge (vertex) connectivity, based on theparameters of graph we obtain sufficient conditions for graphs to be λ3-optimal super-λ3 and super-κ3; For restricted arc connectivity, we use neighborhood con-dition to obtain sufficient conditions for digraphs to be λ′-optimal; We also studythe conditional connectivity of some special class of graphs and discuss the relationbetween inverse degree and connectivity.I 3-restricted edge (vertex) connectivity of graphsStudying the relation between diametergirthdegree sequenceneighbor-hood condition and 3-restricted edge connectivity, We obtain sufficient conditionsfor graphs to be λ3-optimal; Using neighborhood condition  degree sequence degree sum, we obtain sufficient conditions for triangle-free graphs to be super-λ3;Using regular degree, we obtain sufficient conditions for regular graphs to be super-λ3; Using diameter and girth of graph and line graph, we obtain sufficient conditionsfor graphs to be super-λ3 and super-κ3.II Restricted arc connectivity of digraphsLet D be a strongly connected digraph. A strongly connected digraph D iscalled λ′-connected, if restricted arc cuts exist. If xy is an arc with yx /∈ A(D),then call ξ′(xy) = min{d+(x)+ d+(y)− 1, d−(x)+ d−(y)− 1, d+(y)+ d−(x), d+(x)+d−(y) − 1} the arc degree of xy. If xy is an arc with yx ∈ A(D), then call ξ′(xy) =min{d+(x)+d+(y)−2, d−(x)+d−(y)−2, d+(y)+d−(x)−1, d+(x)+d−(y)−1} the arcdegree of xy. The minimum arc degree of D is ξ′(D) = min{ξ′(xy) : xy ∈ A(D)}.A λ′-connected digraph is called λ′-optimal if λ′(D) = ξ′(D).We use neighborhood condition to obtain sufficient conditions for digraphs tobe λ′-optimal.III Conditional connectivity of special graphsLet G = (V, E), the Mycielskian of G is the graph µ(G) whose the vertex setis V (µ(G)) = V ∪ V ′ ∪ {u}, where V ′ = {x′ : x ∈ V } and edge set E(µ(G) =E ∪ {xy′ : xy ∈ E} ∪ {y′u : y′ ∈ V ′}. The vertex x′ is called the twin of the vertexx (and x is the twin of x′) and the vertex u is called the root of µ(G). For n ≥ 2,厦门大学博硕士论文摘要库W <  P xvµn(G) = µ(µn−1(G)).The total graph Tn on n vertices can be obtained from the complete graph Knby adding a loop to every vertex. We define Dn[G] = G ⊗ Tn.We study super connectivity of µ(G) and obtain necessary and sufficient con-ditions for µ(G) to be super (edge) connectivity; We obtain necessary and sufficientconditions for Dn[G] to be super connectivity; We study the super connectivity ofKronecker product of bipartite graphs and complete graphs, super connectivity ofKronecker product of complete graphscyclespaths and complete graphs, superconnectivity of Kronecker product of split graphsthe powers of cyclethe powersof path and complete graphs, and prove that they are super connected; We studythe conditional connectivity of hypercube and folded hypercube and have the valueof some conditional connectivityIV Inverse degree and connectivityLet inverse degree of graph G with no isolated vertex beR(G) =∑v∈V1d(v).We study the relation between inverse degree and n, δ, ξ to obtain sufficient condi-tions for graphs to be super edge connected and λ′-optimal.厦门大学博硕士论文摘要库xvi  /MoM "M:z< Ml . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixuMl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiii<n ti§1.1 WRqza?oZOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.2 KjoT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<I A: 3- Vb#=E§2.1 sE.P7 3- T`!:B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§2.2 B}z 3- T`!:B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<( )W(n_:2 -λ3 =A§3.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17§3.2 0 -λ3 :>74M6i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<0 2 -λ3 =:A§4.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25§4.2 joT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28<O ÆuI D :A:2 -λ3 =b\2 -κ3 =b§5.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30§5.2 0 -λ3 :Z0 -κ3 :sG D 7> . . . . . . . . . . . . . . . . . . . . . . . . . 30<Æ yZA:Vb^=E§6.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39§6.2 λ′- v>74M6i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39< |2-A:=b§7.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47§7.2 > G 7 Mycielskian $_70:` . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47§7.3 Dn[G] :` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50< Kronecker eA:=b§8.1 rg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55§8.2 G*>ZC%>7 Kronecker 70:` . . . . . . . . . . . . . . . . . . . . . . 55§8.3 C%>#ZC%>7 Kronecker 70:` . . . . . . . . . . . . . 58§8.4 M	>#77ZC%>7 Kronecker 70:` . . . . 61<w 2 L5\>2 L5:9l=b厦门大学博硕士论文摘要库Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details.厦门大学博硕士论文摘要库

The Conditional Connectivity of Graphs

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The Conditional Connectivity of Graphs

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