Hedge Connectivity without Hedge Overlaps
HHedge Connectivity without Hedge Overlaps
Rupei Xu and Warren Shull
The University of Texas at DallasEmory University [email protected]@emory.edu
Abstract.
Connectivity is a central notion of graph theory and plays animportant role in graph algorithm design and applications. With emerg-ing new applications in networks, a new type of graph connectivity prob-lem has been getting more attention–hedge connectivity. In this paper,we consider the model of hedge graphs without hedge overlaps, whereedges are partitioned into subsets called hedges that fail together. Thehedge connectivity of a graph is the minimum number of hedges whoseremoval disconnects the graph. This model is more general than the hy-pergraph, which brings new computational challenges. It has been a longopen problem whether this problem is solvable in polynomial time. Inthis paper, we study the combinatorial properties of hedge graph con-nectivity without hedge overlaps, based on its extremal conditions aswell as hedge contraction operations, which provide new insights into itsalgorithmic progress.
Keywords:
Hedge Graph, Graph Connectivity, Graph Contraction
Connectivity has been a central notion of graph theory since its birth in the18th century and has been playing an important role in graph algorithm designand applications. With emerging real-world new applications in image segmen-tation, optical networking, network security, software-defined networking andvirtual network embedding, a new type of graph connectivity problem has beengetting more attention–hedge connectivity, where edges are partitioned into sub-sets called hedges that fail together. The hedge connectivity of a graph is theminimum number of hedges whose removal disconnects the graph. However, itsmathematical studies are still very limited. This is the first paper to investigatethe combinatorial properties of hedge graph connectivity without hedge overlaps,based on its extremal conditions as well as hedge contraction operations.In this paper, all original graphs considered are finite and simple, i.e., theyhave no self-loops nor multiple edges. But during operations, self-loops and mul-tiple edges may appear. Graphs are also assumed connected, otherwise, the hedgeconnectivity is just simply zero.Given an undirected graph G = ( V, E, L ), where V = { v , v , ..., v n } is thevertex set, E = { e , e , ..., e m } is the edge set and L = { (cid:96) , (cid:96) , ..., (cid:96) | L | } is the a r X i v : . [ c s . D M ] D ec Rupei Xu, Warren Shull label set. Each edge has one label from L , and the edge set with same label L i form an hedge H i . Let span ( H i ) represent the span of hedge H i , i.e., thenumber of its components. The rank of a graph rank ( G ) is the difference of thenumber of its vertices and its span: rank ( G ) = | V ( G ) | − span ( G ) . The nullity of a graph nullity is the difference of the number of its edges and its rank: nullity ( G ) = | E ( G ) | − rank ( G ) = | E ( G ) | − | V ( G ) | + span ( G ) . Fig. 1.
Hedge Graph: each edge has a label L j and component index (i). The global hedge connectivity problem asks for finding the minimumnumber of hedges, whose edges removal disconnects the graph. Such set of hedgesis called minimum global hedge cut .Unlike the graph edges adjacency, hedges may have several components. i.e,the span of a hedge is more than 1 . The hedge component-wise adjacent relation-ship can be represented as a square matrix M of order S, where S is the numberof the maximum span of all hedges: S := max pi =1 ( span ( H i )) . For hedge H r andhedge H t , M ij = 1 if and only if the the i -th component of hedge H r and the j -thcomponent of hedge H t intersect each other. The hedge adjacency relationshipcan be represented as a three-dimensional tensor, with three dimensions as hedgeindex, component index, and vertex index. Since this paper mainly focuses oncombinatorial aspects, its matrix and tensor properties are not further studied. edge Connectivity without Hedge Overlaps 3 Despite the st -hedge connectivity problem without hedge overlaps is NP-hard proved by Cai et al.[7], the (global) hedge connectivity problem withouthedge overlaps is polynomial-time solvable in several special cases, includinggraphs with bounded treewidth, planar graphs, and instances with bounded labelfrequency showed by Zhang [6]. It is also in P when the graph has bounded degreeand when for each label and the subgraph induced by the label is connected byCoudert et al.[1], i.e., the span ( H i ) = 1, this problem is equivalent to hypergraphconnectivity problem, which is also known polynomial-time solvable. Xu [5] gavea randomized polynomial-time algorithm for hedgegraphs with constant span.Fox, Panigrahi and Zhang [3] further improved the running time of randomizedpolynomial-time algorithm for hedgegraphs with constant span and hedgegraph- k -cut problem, where k is a constant. The problem is fixed-parameter tractable(FPT) when parameterized by the number k of labels for which the subgraphinduced by the label is not connected by Coudert et al.[2]. It is also known hedgeconnectivity problem without hedge overlaps is quasi-polynomial time solvableby Mohsen, Karger and Panigrahi [4].There are several other easy verifiable polynomial-time solvable cases: (1).If there is only one label, this graph is 1- connected; (2). If there is one vertexhas label degree 1 , the hedge graph is 1-connected; (3). If the number of labelsis constant, it is polynomial-time solvable; (4). If there are m labels, where m is the number of edges, this problem is equivalent to the ordinary graph edgeconnectivity problem. Thus the open case of hedge connectivity without hedgeoverlaps is the following: when there exists a hedge with span no less than 2 , allvertices have label degree no less than 2 , the number of labels is no more than m − , but this number is not a constant.In the following sections, the relationship between hedge connectivity andlabel degree, hedge adjacency and hedge contraction operations are carefullyinvestigated. Let d L ( v ) be the label degree of vertex v ∈ V , which is the number of differ-ent labels on the edges incident with vertex v . Let δ L ( V ) and ∆ L ( V ) be the minimum label degree and maximum label degree of graph G : δ L ( V ) := min v i ∈ V ( G ) { d L ( v i ) } ,∆ L ( V ) := max v i ∈ V ( G ) { d L ( v i ) } . For each hedge H i , let its hedge total label degree total d L ( V ( H i )) be thesum of induced label degrees of all its vertices, δ L ( V ( H i )) and ∆ L ( V ( H i )) beits minimum hedge label degree and maximum hedge label degree , whichare the minimum and maximum value of induced label degrees of all its vertices. total d L ( V ( H i )) := (cid:88) v j ∈ V ( H i ) d L ( v j ) , Rupei Xu, Warren Shull δ L ( V ( H i )) := min v j ∈ V ( H i ) d L ( v j ) ,∆ L ( V ( H i )) := max v j ∈ V ( H i ) d L ( v j ) . It is obvious that δ L ( V ) = min i δ L ( V ( H i )) ,∆ L ( V ) = max i ∆ L ( V ( H i )) . Theorem 1.
The Global Hedge Connectivity (Minimum Global Hedge Cut) ofa graph is at most the minimum label degree of the graph: λ H ( G ) ≤ δ L ( V ) . Proof.
Let L ∗ be the minimum global hedge cut, if | L ∗ | is greater than δ L ( V ),the hedges adjacent to the vertex of minimum label degree can be seen as ahedge cut, as their removal can disconnect the graph, but their size is smallerthan the | L ∗ | , which conflicts L ∗ is the minimum global hedge cut. Theorem 2.
Relabel each hedge with a new label, such that if two hedges adja-cent to each other, the new labels of them are different. The new labels form anew label set L (cid:48) = { L (cid:48) , L (cid:48) , ..., L (cid:48) q } , its size is at least the maximum label degreeof the original graph: | L (cid:48) | ≥ ∆ L ( V ) . Proof.
Assume d L ( v ) = ∆ L ( V ) , there are ∆ L ( V ) hedge edges connected to ver-tex v. If | L (cid:48) | < ∆ L ( V ) , according to pigeonhole principle, there exists two hedgeswhose edges connected to vertex v have the same new label, which contradictswith the relabel rule. Fig. 2.
Hedge Relabelingedge Connectivity without Hedge Overlaps 5
Fig. 3.
Hedge Adjacency with Relabeling
Fig. 4.
Hedge Component-wise Adjacency Rupei Xu, Warren Shull
Let d A ( H i ) be the hedge adjacency degree of hedge H i , which is the numberof other hedges adjacent to hedge H i . Theorem 3. d A ( H i ) ≤ total d L ( V ( H i )) . Proof.
Accoring to the definition, d A ( H i ) is the number of other hedges thathedge H i adjacent to, which is the union of adjacent hedges of all vertices in H i , and its size is no more than the the total number of adjacent hedges of allvertices in H i . Theorem 4. max i d A ( H i ) ≥ ∆ L ( V ) . Proof.
Assume vertex v has the maximum label degree: d L ( v ) = ∆ L ( V ) , and d A ( H k ) = max i d A ( H i ) . If v ∈ V ( H k ) , d A ( H k ) is the union of adjacent hedges ofall vertices in H k , thus max i d A ( H i ) ≥ ∆ L ( V ) . Otherwise, if v ∈ V ( H j ) , where d A ( H j ) ≤ d A ( H k ) , because d L ( v ) ≤ d A ( H j ) , one can get d L ( v ) = ∆ L ( V ) ≤ d A ( H j ) ≤ d A ( H k ) = max i d A ( H i ) . Theorem 5.
Relabel each hedge with a new label, such that if two hedges adja-cent to each other, the new labels of them are different. The new labels form anew label set L (cid:48) = { L (cid:48) , L (cid:48) , ..., L (cid:48) q } , its size is at least the maximum adjacencydegree of the original graph. | L (cid:48) | ≥ max i d A ( H i ) . Proof.
Draw the hedge adjacency graph G A ( V A , E A ) such that each vertex in V A is a hedge of original hedge graph G, two vertices in V A are connected by anedge if and only if the two corresponding two hedges in the original hedge graphare adjacent to each other. This procedure can be done in polynomial-time.Assume | L (cid:48) | < max i d A ( H i ) , and d A ( H k ) = max i d A ( H i ) . According topigeonhole principle, among the hedges adjacent to H k , there are two hedgeshave the same label, which contradicts with the relabel rule. Corollary 1. λ H ( G ) ≤ δ L ( V ) } ≤ ∆ L ( V ) } ≤ max i d A ( H i ) ≤ | L (cid:48) | . Lemma 1. (Vizing’s Theorem)
The number of labels needed to relabel thehedge graph, such that adjacent hedges in the original graph now have differentlabels, is either max i d A ( H i ) or max i d A ( H i ) + 1 . For hedge H i , G (cid:30) H i denotes the graph obtained from G by contracting eachedge e ∈ H i into a single vertex and deleting resulting loops with label of H i . Note that, other loops with different labels from H i caused during contractionof H i , must be kept, unless they can be cleaned up in the following clean-upprocess: (1) merge edges with same labels between two vertices; (2) merge loopswith sames labels of edges on each vertex. edge Connectivity without Hedge Overlaps 7 In contraction of H i , it is easy to verify that, in graph G, the number of edgesis reduced by | H i | , i.e., the number of edges in H i , and the number of vertices isreduced by ( | ( V ( H i ) | − span ( H i )) , i.e., the rank of H i . The rank of H i is reducedto be zero and the nullity of H i is reduced by one; the rank of G is reduced bythe rank of H i and the nullity of G is reduced by the nullity of H i . Theorem 6. rank ( G ) = | L | (cid:88) i rank ( H i ) . Proof.
According to the definition, the rank of graph G is the difference of itsnumber of vertices and the number of its components: rank ( G ) = n − . Inoriginal graph G, the number of vertices is n, after contacting all hedges, thereis only one single vertex left, the number of vertices is reduced by ( n −
1) intotal. In contraction of H i , the number of vertices of graph G is reduced by therank of H i , i.e., | V ( H i ) | − span ( H i ) . Thus, | L | (cid:88) i rank ( H i ) = | L | (cid:88) i ( | V ( H i ) | − span ( H i )) = n − rank ( G ) . Theorem 7. nullity ( G ) = | L | (cid:88) i nullity ( H i ) . Proof.
According to the definition, the nullity of graph G is the difference of thenumber of its edges and its rank: nullity ( G ) = m − n + 1 . From Theorem 6, (cid:80) | L | i nullity ( H i ) = (cid:80) | L | i | H i | − (cid:80) | L | i rank ( H i ) = m − n + 1 = nullity ( G ) . Theorem 8. | L | (cid:88) i =1 | V ( H i ) | = n (cid:88) j =1 d GL ( v i ) . Proof.
Since all hedge vertices cover graph G vertices, at each vertex v i ∈ V ( G ) , it is covered by at least d GL ( v i ) times, thus (cid:80) | L | i =1 | V ( H i ) | ≥ (cid:80) nj =1 d GL ( v i ) . On theother hand, if one hedge vertex v i ∈ H j is contained in a subset of vertices of V ( G ), if the subset of vertices of V ( G ) incident to at least one edge with the samelabel as hedge H j , the label degree all all vertices of G is no less than the union ofall incident edge sets of all subsets of V ( G ) , thus (cid:80) | L | i =1 | V ( H i ) | ≤ (cid:80) nj =1 d GL ( v i ) . Therefore, the equality holds.
Theorem 9. | L | (cid:88) i =1 span ( H i ) ≤ m − n + 1 .nδ L ( v i ) − n + 1 ≤ | L | (cid:88) i =1 span ( H i ) ≤ n∆ L ( v i ) − n + 1 . Rupei Xu, Warren Shull
Proof.
According to Theorem 6, (cid:80) | L | i ( | V ( H i ) | − span ( H i )) = n − , thus (cid:80) | L | i span ( H i ) = (cid:80) | L | i ( | V ( H i ) | ) − n + 1 . According to Theorem 8, (cid:80) | L | i =1 | V ( H i ) | = (cid:80) nj =1 d GL ( v i ) , thus (cid:80) | L | i span ( H i ) = (cid:80) nj =1 d GL ( v i ) − n + 1 . Itis obvious (cid:80) v j ∈ V ( G ) d L ( v j ) ≤ (cid:80) | L | i =1 | H i | = 2 | E | = 2 m, therefore the firstinequality holds. Since δ L ( v i ) ≤ d GL ( v i ) ≤ ∆ L ( v i ) , thus the second inequalityholds. Theorem 10.
Let u and v be two vertices of hedge graph G with label degrees d L ( u ) and d L ( v ) , and e uv ∈ E ( G ) , after contacting the edge e uv between them,the new vertex w replacing them satisfies the following conditions: min { d L ( u ) , d L ( v ) } − ≤ max { d L ( u ) , d L ( v ) } − ≤ d L ( w ) ≤ d L ( u ) + d L ( v ) − ≤ { d L ( u ) , d L ( v ) } − . Proof.
Let L ( E ( u )) and L ( E ( v )) be the labels on edges incident with vertices u and v . After contacting the edge e uv between u and v , the labels on edgesadjacent to the new vertex w are the union of L ( E ( v )) and L ( E ( v )) deducingthe label of the contracted edge, which number is no more than the sum of eachvertex label degree minus 1. d L ( w ) = | ( L ( E ( u )) − L ( e uv )) ∪ ( L ( E ( v )) − L ( e uv )) | ≤ d L ( u ) − d L ( v ) − d L ( u ) + d L ( v ) − . As the number of labels in this union ( L ( E ( u )) − L ( e uv )) ∪ ( L ( E ( v )) − L ( e uv )is no less than the maximum label sets of L ( E ( v )) − L ( e uv ) and L ( E ( v )) − L ( e uv ),thus max { d L ( u ) , d L ( v ) } − ≤ d L ( w ) . Theorem 11. δ L ( V ( G )) − ≤ δ L ( V ( G (cid:30) H i )) . Proof.
Assume v ∈ V ( G ) has the minimum label degree: d L ( v ) = δ L ( V ( G )) . (1) If v / ∈ V ( H i ) , d L ( v ) ≤ δ L ( V ( H i )) , after contracting H i , in worst case, ac-cording to Theorem 10, the new created vertices have minimum label degreeof δ L ( V ( H i )) − . Vertices not in H i keep their original label degrees. Since d L ( v ) ≤ δ L ( V ( H i )) , therefore δ L ( V ( G )) − ≤ δ L ( V ( G (cid:30) H i )) . (2)If v ∈ V ( H i ) ,d L ( v ) = δ L ( V ( H i )) , after contracting H i , according to Theorem 10, the new cre-ated vertices have minimum label degree of δ L ( V ( H i )) − d L ( v ) − . Verticesnot in H i keep their original label degrees. Therefore, δ L ( V ( G )) − ≤ δ L ( G (cid:30) H i ) . Theorem 12. (cid:88) v j ∈ V ( G/H i ) d L ( v j ) ≤ (cid:88) v j ∈ V ( G ) d L ( v j ) − rank ( H i ) . Proof.
According to Theorem 10, after contacting one edge between u and v ,the new vertex w replaced them has label degree d L ( w ) ≤ d L ( u ) + d L ( v ) − . Apply contraction to all edges of hedge H i , for vertices not in V ( H i ) , their labeldegrees keep the same, for vertices in V ( H i ) , their total label degrees are thesize of the union of labels on E ( V ( H i )) deducing the label of H i , which is nomore than the total label degrees of V ( H i ) deducing two times the rank of H i , since the number of vertices reduces by rank ( H i ) , in each operation of mergingtwo vertices, the number of labels in the new created vertex reduces by at least2. edge Connectivity without Hedge Overlaps 9 Theorem 13. (cid:88) v j ∈ V ( G ) d L ( v j ) ≤ (cid:88) v j ∈ V ( G (cid:30) H i ) d L ( v j ) + (cid:88) v j ∈ V ( H i ) d L ( v j ) − span ( H i )( δ ( V ( G )) − . Proof. (cid:80) v j ∈ V ( G ) d L ( v j ) − (cid:80) v j ∈ V ( H i ) d L ( v j ) is the total label degrees of verticesnot in V ( H i ) , after contraction of H i , those total label degrees do not change. (cid:80) v j ∈ V ( G (cid:30) H i ) d L ( v j ) are the total label degrees of all vertices in G/H i , whichcontain the total label degrees of vertices not in V ( H i ) , and the the label degreesof new created vertices after contraction of H i . According to Theorem 11, thetotal label degrees of new created vertices after contraction of H i are no lessthan span ( H i )( δ ( V ( G )) − Theorem 14. d G/H i A ( H j ) = (cid:40) d GA ( H j ) + d GA ( H i ) − | L (cid:48) | + 1 , if H i is adjacent to H j in G ; d GA ( H j ) , otherwise. where | L (cid:48) | is either max i d GA ( H i ) or max i d GA ( H i ) + 1 . Proof. If H i is adjacent to H j in G , after contracting of H i , H i is deleted fromthe adjacency hedge list of H j , but hedges originally adjacent to H i but not H j now become adjacent to H j via the contraction operation. In the relabelprocess, hedges have different new labels if they are adjacent to each other,thus there are ( | L (cid:48) | −
2) number of hedges are adjacent to both H i and H j , thenumber of hedges adjacent to H i but not H j is ( d GA ( H i ) − | L (cid:48) | + 2) . Therefore, d G/H i A ( H j ) = d GA ( H j ) − d GA ( H i ) − | L (cid:48) | + 2) . According to Lemma 1, | L (cid:48) | is either max i d GA ( H i ) or max i d GA ( H i ) + 1 . If H i is not adjacent to H j in G , after contraction of H i , d GA ( H j ) does notchange. Acknowledgments
Rupei Xu would like to express her sincere gratitudes toGuoli Ding and Jie Han for their helpful discussions.
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