NETWORKS of dynamical systems appear in a variety of

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1 Necessary and Suffcent Topologcal Condtons for Identfablty of Dynamcal Networks Henk J. van Waarde, Petro Tes, and M. Kanat Camlbel arxv:807.09v [math.oc] 2 Jul 208 Abstract Ths paper deals wth dynamcal networks for whch the relatons between node sgnals are descrbed by proper transfer functons and external sgnals can nfluence each of the node sgnals. We are nterested n graph-theoretc condtons for dentfablty of such dynamcal networks, where we assume that only a subset of nodes s measured but the underlyng graph structure of the network s known. Ths problem has recently been studed n the case of generc dentfablty. Loosely speakng, generc dentfablty means that the transfer functons n the network can be dentfed for almost all network matrces assocated wth the graph. In ths paper, we nvestgate the stronger noton of dentfablty for all network matrces. To ths end, we ntroduce a new graph-theoretc concept called the graph smplfcaton process. Based on ths process, we provde necessary and suffcent topologcal condtons for dentfablty. Importantly, we also show that these condtons can be verfed by polynomal tme algorthms. Fnally, we show that our results sgnfcantly generalze exstng suffcent topologcal condtons for dentfablty. Index Terms Network Analyss and Control, System Identfcaton, Lnear Systems. I. INTRODUCTION NETWORKS of dynamcal systems appear n a varety of domans, ncludng power systems, robotc networks, and aerospace systems []. In ths paper, we consder dynamcal networks for whch the relatons between node sgnals are modelled by proper transfer functons and external sgnals can nfluence each of the node sgnals. Such network models have receved much attenton (see, e.g., [2] [6]). The nterconnecton structure of a dynamcal network can be represented by a drected graph, where vertces (or nodes) represent scalar sgnals, and edges correspond to transfer functons connectng dfferent node sgnals. We wll assume that the underlyng graph (.e., the topology) of the dynamcal network s known. We remark that methods for topology dentfcaton have also been studed (see, e.g., [7] [2]). In ths paper, we are nterested n condtons for dentfablty of dynamcal networks. Loosely speakng, dentfablty comprses the ablty Henk van Waarde s wth the Bernoull Insttute for Mathematcs, Computer Scence and Artfcal Intellgence, Faculty of Scence and Engneerng, Unversty of Gronngen, P.O. Box 07, 9700 AK Gronngen, The Netherlands. Henk van Waarde s also wth the Engneerng and Technology Insttute Gronngen, Faculty of Scence and Engneerng, Unversty of Gronngen, 977 AG Gronngen, The Netherlands. Emal: h.j.van.waarde@rug.nl. Petro Tes s wth the Engneerng and Technology Insttute Gronngen, Faculty of Scence and Engneerng, Unversty of Gronngen, 977 AG Gronngen, The Netherlands. Petro Tes s also wth the Department of Informaton Engneerng, Unversty of Florence, 09 Florence, Italy. Emal: p.tes@rug.nl, petro.tes@unf.t. Kanat Camlbel s wth the Bernoull Insttute for Mathematcs, Computer Scence and Artfcal Intellgence, Faculty of Scence and Engneerng, Unversty of Gronngen, P.O. Box 07, 9700 AK Gronngen, The Netherlands. Emal: m.k.camlbel@rug.nl. to dstngush between certan models on the bass of measured data. We assume that each node of the network s externally excted by a known sgnal, but the node sgnals of only a subset of nodes s measured. Wthn ths setup, we are nterested n two dentfablty problems. Frstly, we want to fnd condtons under whch the transfer functons from a gven node to ts out-neghbours can be dentfed. Secondly, we wonder under whch condtons all transfer functons n the network can be dentfed. In partcular, our am s to fnd graph-theoretc condtons for the above problems. Condtons based on the topology of the network are desrable snce they gve nsght n the types of network structures that allow dentfcaton, and n addton may ad n the selecton of measured nodes. Graphtheoretc methods have also been succesfully appled to assess other system-theoretc propertes lke structural controllablty [] [] and fault detecton [6]. Identfablty of dynamcal networks s an actve research area (see, e.g., [] [6], [7] [2] and the references theren). The papers that are most closely related to the work presented here are [20], [2], [], and [6], n whch dentfablty s also consdered from graph-theoretc perspectve. In [20] and [2], suffcent graph-theoretc condtons for dentfablty have been presented for a class of state-space systems. In the current paper, we study the transfer functon model ntroduced by Van den Hof et al. [2], whch s more general than the (frstorder) state-space systems n [20] and [2] (for a comparson, we refer to [22]). An addtonal advantage of the condtons presented here s that they are both necessary and suffcent for dentfablty. In [], graph-theoretc condtons have been establshed for generc dentfablty. That s, condtons were gven under whch transfer functons n the network can be dentfed for almost all network matrces assocated wth the graph. The authors of [] show that generc dentfablty s equvalent to the exstence of certan vertex-dsjont paths, whch yelds elegant condtons for generc dentfablty that can be checked n polynomal tme. Inspred by the work n [], we are nterested n graphtheoretc condtons for a stronger noton, namely dentfablty for all network matrces assocated wth the graph (a noton often referred to as global dentfablty). Ths problem s motvated by the fact that, although generc dentfablty guarantees dentfablty for almost all network matrces, there are meanngful examples of network matrces that are not contaned n ths set of almost all network matrces. As a consequence, a stuaton may arse n whch the system under consderaton s not dentfable, even though the condtons for generc dentfablty are satsfed (for an example of such a stuaton, we refer to Secton III). On the other hand, f

2 2 the condtons derved n ths paper are satsfed, then t s guaranteed that the network s dentfable for all network matrces assocated wth the graph. The dfference between generc dentfablty and global dentfablty mght seem subtle at frst, however, smlar dfferences n the controllablty lterature have led to completely dfferent graph-theoretc characterzatons. For nstance, generc controllablty was characterzed n terms of so-called maxmal matchngs [], whle t was shown that strong structural controllablty (.e., controllablty for all network matrces) s equvalent to the exstence of zero forcng sets []. As we wll see, also the mathematcal tools used n ths paper to characterze global dentfablty are completely dfferent from those used n [] to characterze generc dentfablty. In a prelmnary verson of ths work [6], we proved a condton for global dentfablty based on so-called constraned vertex-dsjont paths. The current paper s a sgnfcant contrbuton compared to [6] for two reasons. Frst, the graphtheoretc condtons presented here are both necessary and suffcent for global dentfablty, whle the condtons n [6] are only suffcent. Secondly, the methods used n ths paper to derve such condtons are completely dfferent than the ones n [6]. In fact, we wll ntroduce a new graph-theoretc concept whch we call the graph smplfcaton process. Loosely speakng, the dea s to apply operatons on the graph and measured nodes n such a way that checkng dentfablty becomes easy. Ths approach fundamentally dffers from the path-based condtons n [6]. An mportant fact used n our analyss s that the two dentfablty problems dscussed above are equvalent to the left-nvertblty of certan transfer matrces (for all network matrces) [], [6]. Therefore, as our frst contrbuton we state necessary and suffcent graph-theoretc condtons for leftnvertblty of a transfer matrx usng the graph smplfcaton process. Based on ths result, we obtan necessary and suffcent topologcal condtons for dentfablty. Remarkably, we wll also show that these condtons can be checked n polynomal tme. Fnally, we wll compare our approach to [6], and we wll show that the results presented here generalze those n [6]. Ths paper s organzed as follows. In Secton II we dscuss the prelmnares that are used throughout ths paper. Subsequently, n Secton III we state and motvate the problem. Next, n Secton IV we recall rank condtons for dentfablty. Sectons V and VI contan our man results. In Secton V we ntroduce the graph smplfcaton process and show ts relaton to the rank of transfer matrces. Subsequently, n Secton VI we provde graph-theoretc condtons for dentfablty. Our man results are compared to prevous work n Secton VII. Fnally, Secton VIII contans our conclusons. II. PRELIMINARIES We denote the set of natural numbers by N, real numbers by R, and complex numbers byc. Moreover, the set of realm n matrces s denoted by R m n. The n n dentty matrx s denoted by I n. When ts dmenson s clear from the context, we smply wrte I. A. Polynomals For the sake of completeness, we state some basc defntons and results on polynomals. Let x,x 2,...,x n be ndetermnates and x = (x,x 2,...,x n ). A monomal m(x) n x s a product of non-negatve powers of the ndetermnates x,x 2,...,x n, that s, m(x) s of the form m(x) = x a xa xa n n, where a,a 2,...,a n are non-negatve ntegers. The degree of m(x) s defned as the sum a + a a n. A real polynomal p(x) s the weghted sum of monomals n x,.e., p(x) s of the form p(x) = α m (x)+α 2 m 2 (x)+ +α r m r (x), where r N, α R, and m (x) s a monomal n x for =,2,...,r. The real numbers α,α 2,...,α r are called coeffcents of p(x). The degree of p(x) s defned as the maxmum of the degrees of the monomals wth nonzero coeffcents that appear n p(x). We state the followng basc proposton about nonzero real polynomals. Proposton. Consder k nonzero real polynomals p (x), where =,2,...,k and x = (x,x 2,...,x n ). There exsts an x R n such that p ( x) 0 for all =,2,...,k. The proof of Proposton follows smply from nducton on the number of polynomals, and s therefore omtted. Remark. Wthout loss of generalty, we can assume that x n Proposton has only nonzero coordnates. Indeed, by contnuty, f p ( x) 0 for =,2,...,k, there exsts an open ball B( x) around x n whch p (x) 0 for all =,2,...,k and all x B( x). Obvously, ths open ball contans a pont wth only nonzero coordnates. B. Ratonal functons and ratonal matrces Consder a scalar ndetermnate z and a ratonal functon f(z) = p(z) q(z), where p(z) and q(z) are real polynomals and q s nonzero. We say f s nonzero f p s a nonzero polynomal. Moreover, f s called proper f the degree of p(z) s less than or equal to the degree ofq(z). We say f s strctly proper f the degree of p(z) s less than the degree of q(z). An m n matrx A(z) s called ratonal f ts entres are ratonal functons n the ndetermnate z. In addton, A(z) s proper f ts entres are proper ratonal functons n z. We omt the argument z whenever the dependency of A on z s clear from the context. The normal rank of A(z) s defned as max λ C ranka(λ) and denoted by ranka(z), wth slght abuse of notaton. We say A s left-nvertble f ranka = n. We denote the (,j)- th entry of A by A j. Moreover, the j-th column of A s gven by A j. More generally, let M {,2,...,m} and N {,2,...,n}. Then, A M,N denotes the submatrx of A contanng the rows of A ndexed by M and the columns of A ndexed by N. Next, consder the case that A s square,.e., m = n. The determnant of A s denoted by deta, whle the adjugate of A s denoted by adja. A prncpal submatrx of A s a submatrx A M,M, where M {,2,...,m}. The determnant of A M,M s called a prncpal mnor of A. The next basc result on ratonal matrces s stated for future use.

3 Proposton 2. Let A(z) be an m n ratonal matrx and assume that each row of A(z) contans at least one nonzero entry. There exsts a vector b R n such that each entry of A(z)b s a nonzero ratonal functon. The proof of Proposton 2 follows smply from nducton on the number of rows of A(z) and s therefore omtted. C. Graph theory Let G = (V,E) be a drected graph, wth vertex (or node) set V = {,2,...,n} and edge set E V V. The graphs consdered n ths paper are smple,.e., wthout self-loops and wth at most one edge from one node to another. Consder an edge (,j) E. Then (,j) s called an outgong edge of node V andj s called an out-neghbour of V. The set of outneghbours of s denoted by N +. Smlarly,(, j) s called an ncomng edge of j V and node s called an n-neghbour of j. The set of n-neghbours of node j s denoted by Nj. For any subset S = {v,v 2,...,v s } V we defne the s n matrx P(V;S) as P j := f j = v, and P j := 0 otherwse. The complement ofs nv s defned ass c := V\S. Moreover, the cardnalty of S s denoted by S. A path P s a set of edges n G of the formp = {(v,v + ) =,2,...,k} E, where the vertces v,v 2,...,v k+ are dstnct. The vertex v s called a startng node of P, whle v k+ s the end node. The cardnalty of P s called the length of the path. A collecton of paths P,P 2,...,P l s called vertex-dsjont f the paths have no vertex n common, that s, f for all dstnct,j {,2,...,l}, we have that (u,w ) P,(u j,w j ) P j = u,w,u j,w j are dstnct. Let U,W V be dsjont. We say there exsts a path from U to W f there exst vertces u U and w W such that there exsts a path n G wth startng node u and end node w. Smlarly, we say there are m vertex-dsjont paths from U to W f there exst m vertex-dsjont paths n G wth startng nodes n U and end nodes n W. In the case that U W, we say there exst m vertex-dsjont paths from U to W f there are max{0, m U W } vertex-dsjont paths from U \W to W\U. Roughly speakng, ths means that we count paths of length zero from every node n U W to tself. III. PROBLEM STATEMENT AND MOTIVATION Let G = (V,E) be a smple drected graph wth vertex set V = {,2,...,n} and edge set E V V. Followng the setup of [] (see also [2], []), we assocate the followng dynamcal system wth the graph G: w(t) = G(q)w(t)+r(t)+v(t) y(t) = Cw(t). Here w,r, and v are n-dmensonal vectors of node sgnals, known external sgnals, and unknown dsturbances, respectvely. The (measured) output vector y s p-dmensonal, and conssts of the node sgnals of a subset C V of so-called measured nodes, where C = p. Consequently, the matrx C s defned as C := P(V,C). Moreover, q denotes the forward shft operator defned by qw (t) = w (t + ) (and q s () the backward shft operator satsfyng q w (t) = w (t )). Fnally, G(z) s an n n ratonal matrx, called network matrx, satsfyng the followng propertes [2]: P. For all,j V, the entry G j (z) s a proper ratonal (transfer) functon. P2. The functon G j (z) s nonzero f and only f (,j) E. A matrx G(z) that satsfes ths property s sad to be consstent wth the graph G. P. Every prncpal mnor of lm z (I G(z)) s nonzero. Ths mples that the network model () s well-posed n the sense of Defnton 2. of []. Property P s requred for the techncal analyss n ths paper, but only mposes weak restrctons on the matrx G [2]. Remark 2. We focus on the network model () that was orgnally ntroduced n [2]. Note that state-space network models have also receved much attenton (see, e.g., [7] [9], [2], [9]). A state-space model (wth scalar node dynamcs) can be obtaned from () by choosng the nonzero entres of G as frst-order strctly proper functons (for a comparson of models, see [22]). However, the model () also allows (more general) hgher-order transfer functons. A network matrx G(z) satsfyng Propertes P, P2, and P s called admssble. The set of all admssble network matrces s denoted by A(G). For G(z) A(G), we use the shorthand notatont(z;g) := (I G(z)). If the dependence oft onz and G s clear from the context, we smply wrte T. Moreover, usng basc operatons on the sgnals of (), we fnd that the transfer matrx from r to y s gven by CT(z;G). In ths paper, we are nterested n the queston whch transfer functons n G(z) can be unquely dentfed from nput/output data, that s, from the external sgnals r(t) and output sgnals y(t). To ths end, we assume that the graph G = (V,E) s known. Moreover, we assume that the exctaton sgnal r(t) s suffcently rch such that, under sutable assumptons on the dsturbance v(t), the transfer matrx CT(z; G) can be dentfed from {r(t), y(t)}-data (see, e.g., Chapter 8 of [2]). Note that we are not per se nterested n dentfyngct(z;g), but we want to dentfy (a part of) the transfer matrx G(z). Therefore, the queston s whch transfer functons n G(z) can be unquely reconstructed from the transfer matrx CT(z; G). In recent work [], [2] ths queston has been consdered for generc dentfablty. Graph-theoretc condtons were gven under whch a set of transfer functons can be unquely dentfed from CT(z;G) for almost all network matrces G consstent wth the graph. For a formal defnton of generc dentfablty we refer to Defnton of []. Here, we wll nformally llustrate the approach of []. Example. Consder the graph G = (V,E) n Fgure. We assume that the node sgnals of nodes andcan be measured, that s, C = {,}. Suppose that we want to dentfy the transfer functons from node to ts out-neghbours,.e., the transfer functons G 2 (z) and G (z). Accordng to Corollary. of [], ths s possble f and only f there exst two vertexdsjont paths from N + to C. Note that ths s the case n ths example, snce the edges (2,) and (,) are two vertexdsjont paths. To see why we can genercally dentfy the transfer functons G 2 and G, we compute CT as:

4 Fg.. Graph used n Example. CT = 2 ( ) G2 G 2 +G G G 2 G 0, G 2 G 2 +G G G 2 G 0 where we omt the argument z. Clearly, we can unquely dentfy the transfer functons G 2,G,G 2, and G from CT. Moreover, the transfer matrces G 2 and G satsfy ( )( ) ( ) G2 G G2 T =. (2) G 2 G G T Equaton (2) has a unque soluton n the unknowns G 2 and G f G 2 G G G 2 0, whch means that we can dentfy G 2 and G for almost all G consstent wth G. As mentoned before, the approach based on vertex-dsjont paths [] gves necessary and suffcent condtons for generc dentfablty. Ths mples that for some network matrces G, t mght be mpossble to dentfy the transfer functons, even though the path-based condtons are satsfed. For nstance, n Example we cannot dentfy the functons G 2 and G f the network matrx G s such that G 2 = G = G 2 = G. Nonetheless, scenaros n whch some (or all) transfer functons n the network are equal occur frequently, for example n the study of undrected (electrcal) networks [2], n unweghted consensus networks [26], and n the study of Cartesan products of graphs [27]. Therefore, nstead of generc dentfablty, we are nterested n graph-theoretc condtons that guarantee dentfablty for all admssble network matrces. Such a problem mght seem lke a smple extenson of the work on generc dentfablty []. However, to analyze strong structural network propertes (for all network matrces), we typcally need completely dfferent graph-theoretc tools than the ones used n the analyss of generc network propertes. For nstance, n the lterature on controllablty of dynamcal networks, generc controllablty s related to maxmal matchngs [], whle strong structural controllablty s related to zero forcng sets [] and to constraned matchngs [27]. To make the problem of ths paper more precse, we state a few defntons. Frstly, we are nterested n condtons under whch all transfer functons from a node to ts outneghbours N + are dentfable (for any admssble network matrx G A(G),.e., any G that satsfes propertes P, P2, and P). If ths s the case, we say (,N + ) s dentfable. More precsely, we have the followng defnton. Defnton. Consder a drected graph G = (V,E) and let V and C V. Moreover, defne C = P(V,C). We say (,N + ) s dentfable from C f the mplcaton CT(z;G) = CT(z;Ḡ) = G (z) = Ḡ (z) holds for all G(z),Ḡ(z) A(G). In addton to dentfablty of (,N + ), we are nterested n condtons under whch the entre network matrx G can be dentfed from the transfer matrx CT. If ths s the case, we say the graph G s dentfable. Defnton 2. Consder a drected graph G = (V,E) and let C V and C = P(V,C). We say G s dentfable from C f the mplcaton CT(z;G) = CT(z;Ḡ) = G(z) = Ḡ(z) holds for all G(z),Ḡ(z) A(G). The man goal of ths paper s to fnd graph-theoretc condtons for the two notons of dentfablty stated above. We formalze the problem n ths paper as follows. Problem. Consder a drected graph G = (V, E) wth measured nodes C V. Provde necessary and suffcent graph-theoretc condtons under whch, respectvely, (,N + ) and G are dentfable from C. To deal wth Problem, we make use of rank condtons for dentfablty whch we wll recall n the next secton (Secton IV). Subsequently, to verfy such rank condtons, we ntroduce a novel graph-theoretc concept called the graph smplfcaton process (Secton V). The basc dea s that we can perform certan operatons on the graph and on the set of measured nodes that do not affect the rank of certan transfer matrces (and hence, do not affect dentfablty). We wll show that the consecutve applcaton of such operatons yelds a new graph called derved graph and new set of measured nodes called derved vertex set. It turns out that dentfablty n the orgnal graph can be equvalently checked n the derved graph (n whch t s easy to check dentfablty). Ths yelds neat graph-theoretc necessary and suffcent condtons for dentfablty, whch can be verfed by polynomal tme algorthms. In addton, t wll be shown n Secton VII that our results sgnfcantly generalze exstng suffcent condtons for dentfablty based on so-called constraned vertexdsjont paths [6]. IV. RANK CONDITIONS FOR IDENTIFIABILITY Frst, we revew some of the condtons for dentfablty n terms of the normal rank of transfer matrces. For the proofs of all results n ths secton, we refer to [6]. Recall from Secton II that T + C,N (z;g) denotes the submatrx of T formed by takng the rows of T ndexed by C and the columns of T correspondng to N +. The followng lemma (Lemma n [6]) states that dentfablty of (,N + ) s equvalent to a rankcondton on the matrx T + C,N (z;g). Lemma. Consder a drected graph G = (V,E), let V, and C V. Then, (,N + ) s dentfable from C f and only f rankt + C,N (z;g) = N + for all G(z) A(G). As an mmedate consequence of Lemma, we fnd condtons for the dentfablty of G based on the normal rank of transfer matrces. Ths s stated n the followng corollary. Corollary. Consder a drected graph G = (V, E) and let C V. Then, G s dentfable from C f and only f rankt + C,N (z;g) = N + for all V and all G(z) A(G).

5 Although Lemma and Corollary gve necessary and suffcent condtons for the dentfablty of respectvely (,N + ) and G, these condtons are lmted snce there s no obvous method to check left-nvertblty of T + C,N (z;g) for an nfnte number of matrces G. Therefore, one of the man results of ths paper wll be graph-theoretc condtons for the left-nvertblty of T W,U (z;g), where U,W V are any two subsets of vertces. These condtons wll be ntroduced n the next secton. V. THE GRAPH SIMPLIFICATION PROCESS In ths secton we provde necessary and suffcent condtons for left-nvertblty of T W,U (z;g) for all G(z) A(G), where U,W V. Loosely speakng, the dea s to smplfy the graph G and nodes W n such a way that checkng leftnvertblty becomes easy. To gve the reader some ntuton for the approach, we start wth the followng basc lemma, whch asserts that T W,U (z;g) s left-nvertble f U W. Lemma. Consder a drected graph G = (V, E) and let U,W V. If U W then rankt W,U (z;g) = U for all G(z) A(G). The proof of Lemma s postponed to Appendx A snce an elementary proof of ths statement can be gven usng deas that are developed later n ths secton (Lemma 6). The condton U W consdered n Lemma s clearly not necessary for left-nvertblty. One can show ths usng the example G = (V,E), where V = {,2}, E = {(,2)}, and the subsets U and W are chosen as U = {} and W = {2}. However, the man dea of the graph smplfcaton process s to smplfy G and to move the nodes n W closer to the nodes n U such that the condton U W possbly holds after applyng these operatons. Of course, we cannot blndly modfy the graphg snce ths would affect the left-nvertblty of T W,U (z;g). Instead, we wll now state two lemmas n whch we consder two dfferent operatons on G and W that preserve left-nvertblty of T W,U (z;g). We emphasze that the graph operatons are ntroduced for analyss purposes only. Indeed, snce the condton of Lemma s smple to check, the graph operatons should be seen as a tool to check leftnvertblty of the transfer matrx of a gven fxed graph G. Frst, we state Lemma 6 whch asserts that left-nvertblty of T W,U (z;g) s unaffected by the removal of the outgong edges of W. Lemma 6. Consder a drected graph G = (V, E) and let U,W V. Moreover, let Ḡ = (V,Ē) be the graph obtaned from G by removng all outgong edges of the nodes n W. Then rankt W,U (z;g) = U for all G(z) A(G) f and only f rankt W,U (z;ḡ) = U for all Ḡ(z) A(Ḡ). Proof. Let G(z) A(G). Relabel the nodes n V such that ( ) GR,R G G = R,W, () G W,R G W,W where R := V \ W and the argument z has been omtted. Defne the matrx Ḡ as ( ) GR,R 0 Ḡ =. () G W,R 0 The matrx Ḡ s an admssble matrx consstent wth Ḡ,.e., Ḡ A(Ḡ). To see ths, note that Ḡ satsfes Property P. Moreover, snce all outgong edges of nodes nw are removed n the graph Ḡ, the matrx Ḡ s consstent wth Ḡ. Hence, Ḡ satsfes property P2. Fnally, to see that Ḡ satsfes Property P, note that any prncpal mnor of ( ) I GR,R (z) 0 lm () z G W,R (z) I s ether or equal to a prncpal mnor of lm z (I G R,R (z)), whch s nonzero by the assumpton that G s admssble. We conclude that Ḡ A(Ḡ). Next, by Proposton of [28], the nverse of I G can be wrtten as ( ) T = (I G) = S(G)G W,R (I G R,R ), S(G) where S(G) := (I G W,W G W,R (I G R,R ) G R,W )) denotes the nverse Schur complement of I G. Usng the same formula to compute the nverse of I Ḡ, we fnd ( ) T := (I Ḡ) = G W,R (I G R,R ). I The above expressons for T and T mply that T W,U = S(G) T W,U, and because S(G) has full normal rank, we obtan rankt W,U = rank T W,U. (6) Next, we use (6) to prove the lemma. Frst, to prove the f statement, suppose that rankt W,U (z;ḡ) = U for all matrces Ḡ A(Ḡ). Let G A(G). Usng G, construct the matrx Ḡ A(Ḡ) n (). By hypothess, rankt W,U (z;ḡ) = U and therefore we conclude from (6) that rankt W,U (z;g) = U. Subsequently, to prove the only f statement, suppose that rankt W,U (z;g) = U for all G(z) A(G). Consder any matrx Ḡ(z) A(Ḡ) and note that Ḡ can be wrtten n the form (). Next, we choose the matrces G R,W and G W,W such that the matrx G n () s consstent wth the graph G, and such that the nonzero entres of G R,W and G W,W are strctly proper ratonal functons. Ths means that G readly satsfes Propertes P and P2 (see Secton III). In fact, G also satsfes P. Indeed, snce lm z (I G(z)) s gven by (), t follows that every prncpal mnor of lm z (I G(z)) s ether or equal to a prncpal mnor of lm z (I G R,R ), whch s nonzero by the hypothess that Ḡ(z) A(Ḡ). We conclude that G satsfes Propertes P, P2, and P, equvalently, G A(G). By hypothess, rankt W,U (z;g) = U and consequently, by (6) we conclude that rankt W,U (z;ḡ) = U. Ths proves the lemma. Remark. Usng the exact same arguments as n the proof of Lemma 6, we can also prove that all ncomng edges of nodes n U can be removed wthout affectng the left-nvertblty of the matrx T W,U (z;g). Inspred by Lemma 6, we wonder what type of operatons we can further perform on the graph G and nodes W wthout affectng left-nvertblty of T W,U (z;g). In what follows we

6 6 wll show that under sutable condtons t s possble to move the nodes n W closer to the nodes n U. Here the noton of reachablty n graphs wll play an mportant role. For a subset U V and a node j V \U, we say j s reachable from U f there exsts at least one path from U to j. By conventon, f j U then j s reachable from U. In the followng lemma, we wll show that the rank of T W,U (z;g) s unaffected f we replace a node k W \ U by j, provded that j s the only n-neghbour of k that s reachable from U. Lemma 7. Consder a drected graph G = (V, E) and let U,W V. Suppose that k W \ U has exactly one nneghbour j N k that s reachable from U. Then for all G(z) A(G), we have rankt W,U (z;g) = rankt W,U (z;g), where W := (W \{k}) {j}. Remark. We emphasze that Lemma 7 does not requre node k to have exactly one n-neghbour. In general, node k may have multple n-neghbours, but f exactly one of such neghbours s reachable from U, we can apply Lemma 7. In addton, we remark that Lemma 7 s qute ntutve. Indeed, under the assumptons of Lemma 7, all nformaton from the nodes n U enters nodek va nodej. Therefore, choosng node k or node j as a node n W does not make any dfference. An nterestng specal case s obtaned when both nodes j and k are contaned n W. In ths case, we obtan W = W \ {k}, that s, node k can be removed from W wthout affectng the rank of T W,U (z;g). Proof of Lemma 7. By Lemma 6, we can assume wthout loss of generalty that the nodes n W have no outgong edges. Let G(z) A(G). In what follows we omt the dependence of G on z and the dependence of T(z;G) on both z and G. Consder a vertex v U. Note that (I G)T = I (7a) n (I G) kl T lv = 0, (7b) l= where n := V and (7b) follows from the fact that k W\U and v U are dstnct. Equaton (7b) mples that T kv = G kl T lv. (8) l N k Note that j N k, but possbly N k contans other vertces. We wll now prove that for all these other vertces, the correspondng transfer functon T lv equals zero. That s, T lv = 0 for all l N k \{j}. To see ths, we frst observe that there does not exst a path n G from v to l N k \ {j}. Indeed, suppose that there s a path P from v to l. Then ths path cannot contan the edge (j,k), snce node k W \ U does not have any outgong edges. Ths mples that there exsts a path P (l,k) from v to k va node l. Ths s a contradcton snce by hypothess j s the only n-neghbour of k that s reachable from U. Therefore, we conclude that there does not exst a path from v to l. By Lemma of [2] we conclude that T lv = 0. Ths means that (8) can be smplfed as T kv = G kj T jv. Snce v U s arbtrary, t follows that As G kj 0, we conclude that T k,u = G kj T j,u. rankt W,U = rankt W,U, where W := (W \{k}) {j}. Ths proves the lemma. From Lemma 6 and Lemma 7, we see that () we can always remove the outgong edges of nodes n W and () we can move nodes n W closer to U under sutable condtons. Of course, snce both operatons do not affect left-nvertblty of T W,U, we can also apply these operatons multple tmes consecutvely. Therefore, we ntroduce the followng process to smplfy the graph G and move the nodes n W. The dea of ths process s to apply the above operatons to the graph untl no more changes are possble. Graph smplfcaton process: Let G = (V,E) be a drected graph and let U,W V. Consder the followng two operatons on the graph G and nodes W. ) Remove all outgong edges of nodes n W from G. 2) If k W \U has exactly one n-neghbour j N k that s reachable from U, replace k by j n W. Consecutvely apply operatons and 2 on the graph G and nodes W untl no more changes are possble. Clearly, the graph smplfcaton process termnates after a fnte number of applcatons of operatons and 2. Indeed, operaton can only be appled once n a row, and a node n W \ U can be moved at most V tmes whch means that operaton 2 can be appled only fnte number of tmes. In fact, f operatons and 2 are consecutvely appled (n ths order), then the process termnates wthn V operatons of type and 2. Ths s due to the fact that f the outgong edges of a node j V are removed, then we cannot apply operaton 2 to replace a node k by j. A graph obtaned by applyng the graph smplfcaton process to G s called a derved graph, whch we denote by D(G). Smlarly, we call a vertex set obtaned by applyng the graph smplfcaton process to W a derved vertex set, denoted by D(W). To stress the fact that D(G) and D(W) do not only depend on the graph G and set W, but also on the set U, we say that D(G) and D(W) are a derved graph of G and derved vertex set of W wth respect to the set U. We emphasze that derved graphs and derved vertex sets are not necessarly unque. In general, the derved graph and derved vertex set that are obtaned from the graph smplfcaton process depend on the order n whch the operatons and 2 are appled, and on the order n whch operaton 2 s appled to the nodes n W. However, t turns out that the non-unqueness of derved graphs and derved vertex sets s not a problem for the applcaton (left-nvertblty) we have n mnd. In fact, we wll show n Theorem 8 that any

7 7 derved graph and derved vertex set wll lead to the same conclusons about left-nvertblty. We wll llustrate the graph smplfcaton process n Example 2. Remark. In step 2 of the graph smplfcaton process, we have to decde whether there exsts a node k W \ U that has exactly one n-neghbour j N k whch s reachable from U. Therefore, we want to fnd whch n-neghbours of k are reachable from U. One of the ways to do ths, s to use Djkstra s sngle source shortest path (SSSP) algorthm [29], [0]. Ths algorthm computes the shortest paths (.e., paths of mnmum length) from a gven source node s to every other node n the graph, and returns an nfnte dstance for each node whch s not reachable from s. If we apply the SSSP algorthm to each node n U, we obtan all nodes n V that are reachable from U. Djkstra s SSSP algorthm has tme complexty O(n +e), where n = V and e = E [0], and therefore we can fnd all nodes reachable from U n tme complexty O(un +ue), where u = U. Once we know the nodes n V that are reachable from U, we can smply check whether there exsts exactly one j N k that s reachable from U. In partcular, ths shows that the graph smplfcaton process can be mplemented n polynomal tme snce both operatons and 2 can be mplemented n polynomal tme, and the graph smplfcaton process executes at most n operatons of type and 2 (f appled n ths order). Example 2. Consder the graph G = (V,E) n Fgure 2 and defne U := {2} and W := {,6}. The goal of ths example s to apply the graph smplfcaton process to obtan a derved graph and derved vertex set. After ths smplfcaton, t wll be easy to check left-nvertblty of T W,U (z;g). 2 Fg. 2. Graph G wth nodes W colored black. Frst, note that both nodes and 6 do not have outgong edges, so at the moment we cannot apply operaton. Furthermore, note that node has two n-neghbours that are reachable from U. Hence, we cannot apply operaton 2 to node. However, we observe that node 6 has exactly one n-neghbour (node ) that s reachable from U. Consequently, we can replace node 6 by node n W (see Fgure ). 2 Fg.. Graph wth nodes W (n black), obtaned by applyng operaton 2 to node Subsequently, we see that node has outgong edges, whch we can remove by applyng operaton. Ths s depcted n Fgure. 2 Fg.. Graph wth nodes W (n black), obtaned by applyng operaton to node. Next, node has exactly one n-neghbour that s (trvally) reachable from U. Therefore, we replace by 2 n W. Subsequently, we can remove all outgong edges of node 2. These result of these two operatons s depcted n Fgure. 2 Fg.. Derved graph D(G) wth derved vertex set D(W) (n black), obtaned by applyng operaton 2 to node and operaton to node 2. Note that nodes 2 and do not have any outgong edges. Moreover, the n-neghbour of node s not reachable from node 2, so we cannot use operaton 2 to node. In addton, operaton 2 cannot be appled to node 2 snce 2 U. Therefore, the graph smplfcaton process termnates. We conclude that the graph D(G) n Fgure s a derved graph of G, whereas the vertex set D(W) = {2,} s a derved vertex set of W (wth respect to U). Ths example shows the strength of the graph smplfcaton process n the followng way: snce U D(W), we conclude by Lemma that T D(W),U (z;g) s left-nvertble for all G(z) A(D(G)). However, by Lemma 6 and Lemma 7, we mmedately see that T W,U (z;g) s left-nvertble for all G(z) A(G). Ths suggests that the graph smplfcaton process s a promsng tool to study left-nvertblty of transfer matrces (and hence, to study dentfablty of dynamcal networks). To summarze, we have seen that t s possble to remove the outgong edges of nodes n W and to move the nodes n W closer to U f certan condtons are satsfed. Snce leftnvertblty s preserved by both operatons due to Lemmas 6 and 7, we see that left-nvertblty of T W,U (z;g) for all G(z) A(G) s equvalent to the left-nvertblty of T D(W),U (z;g) for all G(z) A(D(G)). Usng Lemma, ths shows that the condton U D(W) s suffcent for the left-nvertblty of T W,U (z;g). Remarkably, the condton U D(W) turns out to be also necessary for left-nvertblty of T W,U (z;g). Ths s stated more formally n the followng theorem, whch s one of the man results of ths paper. 6 6

8 8 Theorem 8. Consder a drected graph G = (V,E) and let U,W V. Let D(W) be any derved vertex set of W wth respect to U. Then rankt W,U (z;g) = U for all matrces G(z) A(G) f and only f U D(W). Before we prove Theorem 8, we need some auxlary results. Consder a drected graph G = (V,E), let n = V, s = E, and ndex the edges ase = {e,e 2,...,e s }. We assocate wth each edge e E an ndetermnate g e. Moreover, we defne the s-dmensonal vector g := ( g e g e2... g es ), whch we call the ndetermnate vector of G. Next, we defne the n n matrx G as { gek f e k = (,j) for some k G j = 0 otherwse. We emphasze that not all entres of G are ndetermnates, but some are fxed zeros. Note that we wrte G n sans-serf font, to clearly dstngush between G and a fxed ratonal matrx G(z). It s clear that the determnants of square submatrces of I G are real polynomals n the ndetermnate entres of G,.e., n the ndetermnate vector g. Hence, the entres of the adjugate of I G are real polynomals n g. We state the followng basc lemma, whch gves condtons under whch an entry of adj(i G) s a nonzero polynomal. Lemma 9. Consder a drected graph G = (V, E) and let,j V. Let g and G be the ndetermnate vector and matrx of G, respectvely, and defne A := adj(i G). Then A j s a nonzero polynomal n g f and only f there exsts a path from to j. Lemma 9 follows from Proposton. of []. Wth ths lemma n place, we are ready to prove Theorem 8. Proof of Theorem 8. Let D(G) and D(W) be a derved graph and derved vertex set wth respect to U obtaned from the graph smplfcaton process. To prove the f statement, suppose that U D(W). By Corollary we fnd that rankt D(W),U (z;g) = U for all G(z) A(D(G)). By consecutve applcaton of Lemmas 6 and 7, we conclude that rankt W,U (z;g) = U for all G(z) A(G). Conversely, to prove the only f statement, suppose that U D(W). We want to prove that rankt D(W),U (z;g) < U for some G(z) A(D(G)). Snce U D(W), the set Ū := U \ D(W) s nonempty. Furthermore, as D(G) and D(W) result from the graph smplfcaton process, t s clear that nodes n D(W) do not have outgong edges. In addton, each node n the set W := D(W)\U has ether zero or at least two n-neghbours that are reachable from U. As nodes n D(W) U have no outgong edges, ths means that each node n W has ether zero or at least two n-neghbours that are reachable fromū. Fnally, we assume that the nodes n U do not have any ncomng edges, whch s wthout loss of generalty by Remark. The dea of the proof s to show that T D(W), Ū(z;G)b = 0, for some to-be-determned network matrx G(z) A(D(G)) and real vector b. Consequently, rankt D(W), Ū(z;G) < Ū and snce T D(W), Ū s a submatrx of T D(W),U, t wll then mmedately follow that rankt D(W),U (z;g) < U. We nvestgate a row T w, Ū(z;G) of the transfer matrx T D(W), Ū(z;G) and we dstngush two cases, namely the case that w D(W) U and the case that w W. Frst, suppose that w D(W) U. Ths mples that w U. Recall that we assumed wthout loss of generalty that the nodes n U do not have any ncomng edges. Consequently, there are no paths from v to w for any v Ū. We conclude from Lemma of [2] that T wv (z;g) = 0 for all G(z) A(D(G)). Therefore, T w, Ū(z;G) = 0 for all G(z) A(D(G)). Obvously, ths mples that T w, Ū(z;G)b = 0 for all G(z) A(D(G)) and all real vectors b. Next, we consder the second case n whch w W. Let G denote the ndetermnate matrx of D(G). Defne A := adj(i G). Then, we have (I G)A = det(i G)I (9a) (I G) W,V A V, Ū = 0, (9b) where (9b) follows from the fact that Ū and W are dsjont. Recall that nodes n W do not have any outgong edges, and therefore (I G) W, W = I. Ths means that we can rewrte (9b) as A W, Ū = G W, Wc A Wc, Ū, (0) where we recall that Wc := V\ W. Note that for j W c, the column G W,j s equal to 0 f j s not an n-neghbour of any node n W. In addton, for any j W c, the row A j, Ū equals 0 f there s no path from Ū to j (by Lemma 9). Therefore, we can rewrte (0) as A W, Ū = G W,N A N, Ū, () where N W c s characterzed by the followng property: we have j N f and only f j s an n-neghbour of a node n W and there s a path from Ū to j. By defnton of the adjugate, the entres of A N, Ū are polynomals n the ndetermnate entres of G. We clam that the ndetermnate entres of G W,N do not appear n any entry of A N, Ū, that s, A N, Ū s ndependent of the ndetermnate entres of G W,N. For the sake of clarty, we postpone the proof of ths clam to the end. For now, we assume that A N, Ū s ndependent of the ndetermnate entres of G W,N. We recall that there s a path from Ū to each node n N. Let N = {n,n 2,...,n r }, where r = N. Then, for each node n N, there exsts a node u Ū such that A n,u s a nonzero polynomal n the ndetermnate entres of G (by Lemma 9). We emphasze that u and u j are not necessarly dstnct. We focus on the r nonzero polynomals A n,u,a n2,u 2,...,A nr,u r. (2) The dea s to apply Proposton and Remark to these r polynomals. By Remark, we can substtute nonzero real numbers for the ndetermnate entres of G such that all r polynomals (2) evaluate to nonzero real numbers. Snce the polynomals (2) are ndependent of the ndetermnate entres of G W,N, we do not have to fx the entres of G W,N. In add-

9 9 ton, t s possble to substtute strctly proper functons nz for the ndetermnate entres of G (except for entres of G W,N ) such that the polynomals (2) evaluate to nonzero ratonal functons. Indeed, one can smply choose all ndetermnate entres of G (except for the entres of G W,N ) as nonzero real numbers as before, and then dvde all of these real numbers by z. To summarze the progress so far, we have substtuted strctly proper functons for the ndetermnate entres of G (except for the entres ofg W,N ) such that the polynomals (2) evaluate to nonzero ratonal functons. Note that ths mples that the matrx A N, Ū evaluates to a ratonal matrx, whch we denote by A N, Ū(z) from now on. Snce each row of A N, Ū(z) contans a nonzero ratonal functon, by Proposton 2 there exsts a nonzero real vector b such that A N, Ū(z)b has only nonzero ratonal entres. Subsequently, we wll choose the ndetermnate entres of G W,N such that G W,N A N, Ū(z)b = 0. Recall that the nodes n W ether have zero or at least two n-neghbours from the set N. If a node w W has no n-neghbours, then G w,n = 0, and therefore clearly G w,n A N, Ū(z)b = 0. If a node w W has at least two n-neghbours, say n,n 2,...,n p N, then we substtute strctly proper functons for the ndetermnate entres G w,n,g w,n2,...,g w,np so that G w,n A N, Ū(z)b = 0. Note that ths s possble snce the vector A N, Ū(z)b has only nonzero ratonal entres. To conclude, we have substtuted strctly proper functons for the ndetermnate entres of G whch yelds a matrx whch we denote by G(z). The adjugate of I G(z) s denoted by A(z) = adj(i G(z)). We have shown that G W,N (z)a N, Ū(z)b = 0. By (), ths yelds A W, Ū(z)b = 0. Note that det(i G(z)) s nonzero snce all nonzero entres of G are strctly proper functons. Therefore, we can express T(z;G) as T(z;G) = det(i G(z)) A(z), from whch we fnd that T W, Ū(z;G)b = 0. Consequently, T D(W), Ū(z;G)b = 0, and rankt D(W), Ū(z;G) < Ū. Therefore, we conclude that rankt D(W),U (z;g) < U. We stll have to show that G(z) s admssble,.e., G(z) A(D(G)). Snce the ndetermnate matrx G s consstent wth the graph D(G) and we substtuted (nonzero) strctly proper functons for each ndetermnate entry of G, the matrx G(z) readly satsfes Propertes P and P2. In addton, snce all nonzero entres of G(z) are strctly proper, we obtan lm I G(z) = I, z and hence, G(z) also satsfes Property P. We conclude that rankt D(W),U (z;g) < U for some G(z) A(D(G)). Fnally, by consecutve applcaton of Lemmas 6 and 7, we conclude that rankt W,U (z;g) < U for some G(z) A(G). Fnally, recall that we have so far assumed that A N, Ū s ndependent of the ndetermnate entres of G W,N. It remans to be shown that ths s true. To ths end, label the nodes n V such that G can be wrtten as ( ) G G = Wc, W c G Wc, W (a) G W, G Wc W, W ( ) G = Wc, 0 Wc, (b) G W, 0 Wc where (b) follows from the fact that nodes n outgong edges. Ths mples that ( ) I G I G = Wc, 0 Wc, I G W, Wc W have no and therefore ( ) adj(i G A = adj(i G) = Wc, W c ) 0. () Snce the entres of G Wc, Wc are ndependent of the ndetermnate entres of G W, Wc, we conclude from () that the matrx A Wc, W c = adj(i G Wc, W c ) s ndependent of the ndetermnate entres of G W, Wc. Now, to prove the clam, note thatū and W are dsjont by defnton, and thereforeū W c. In addton, we have N W c. Therefore, the matrx A N, Ū s a submatrx of A Wc, Wc. Furthermore, we see that G W,N s a submatrx of G W, Wc by usng the fact that N W c. We conclude that the entres of the matrx A N, Ū are ndependent of the ndetermnate entres of G W,N, whch completes the proof. VI. IDENTIFIABILITY AND GRAPH SIMPLIFICATION In ths secton we use Theorem 8 to provde solutons to the dentfablty problems ntroduced n Secton III. Specfcally, the followng theorem states necessary and suffcent graphtheoretc condtons for dentfablty of (,N + ). Theorem 0. Consder a drected graph G = (V, E), let V and C V. Moreover, let D(C) be any derved vertex set of C wth respect to N +. Then (,N + ) s dentfable from C n G f and only f N + D(C). Theorem 0 follows from Theorem 8 and Lemma. The next result gves necessary and suffcent graph-theoretc condtons under whch the entre graph G s dentfable. Ths result s a corollary of Theorem 0 but s stated as a theorem due to ts mportance. Theorem. Consder a drected graph G = (V,E) and let C V. Then G s dentfable from C f and only f for all V, we have N + D(C), where D(C) s any derved vertex set of C wth respect to N + We emphasze that the derved set D(C) of C depends on the choce of neghbour set N +, and hence, for each node V we have to compute the derved set of C wth respect to N +. To llustrate Theorem, we consder the followng example. Example. Consder the graph G = (V, E) n Fgure 6. Suppose that the measured nodes are gven byc = {,6,7}. In ths example, we want to check whether G s dentfable from C,.e., we want to check whether all transfer functons appearng n the matrx G(z) can be dentfed for any G(z) A(G)..

10 0 Fg. 6. Graph wth measured nodes C = {,6,7} colored black. 2 Followng Theorem, for each V we have to check whether the condton N + D(C) s met, where D(C) s any derved set ofc wth respect ton +. We start wth node. Note that N + = {2,,}. The derved set of C wth respect N + s D(C) = {2,,}. Indeed, ths can be shown by applyng the followng operatons consecutvely: remove the outgong edges of 6 usng operaton, replace measured node 7 by node usng operaton 2, remove outgong edges of node usng operaton, replace node 6 by node usng operaton 2, remove outgong edges of node usng operaton, replace node by node 2 usng operaton 2, and fnally remove the outgong edges of node 2 usng operaton. Ths shows that (,N + ) s dentfable. Secondly, we consder node 2. Note that N 2 + = {,,}. Ths tme, we apply the followng operatons to compute the derved set of C wth respect to N 2 + : remove outgong edges of node 6, replace node 7 by node, remove outgong edges of node, replace node by node, replace node 6 by node, and fnally remove the outgong edges of nodes and. Ths shows that the derved set of C wth respect to N 2 + s equal to {,,} = N 2 +, whch mples that (2,N 2 + ) s dentfable. The dentfablty of (,N + ) s easy to check snce a derved vertex set of C wth respect to N + s smply gven by {,6,7} (obtaned by removng the outgong edges of 6). As N + = {,6} {,6,7}, we conclude that (,N + ) s dentfable. Smlarly, we conclude that (6,N 6 + ) s dentfable. Moreover, snce nodes and 7 do not have outneghbours, we do not have to check dentfablty of (,N + ) and (7,N 7 + ). Therefore, t remans to be shown that (,N + ) s dentfable. Note that N + = {,6,7}. We can apply the followng operatons to obtan the derved set ofc wth respect to N + : remove the outgong edges of 6, replace by, and fnally remove the outgong edges of. Ths shows that the derved set of C wth respect to N + s {,6,7} = N +, whch proves that (,N + ) s dentfable. To conclude, we have shown that (,N + ) s dentfable for all V whch mples that G s dentfable from C. VII. COMPARISON TO RESULTS BASED ON CONSTRAINED VERTEX-DISJOINT PATHS In the prevous secton we establshed necessary and suffcent graph-theoretc condtons for the dentfablty of respectvely(,n + ) and G. The purpose of the current secton s to compare these results to the ones based on so-called 7 6 constraned vertex-dsjont paths [6]. Such paths were used n [6] to provde graph-theoretc condtons for dentfablty. Frst, we recall the defnton of constraned vertex-dsjont paths. Defnton. Let G = (V,E) be a drected graph. Consder a set of m vertex-dsjont paths n G wth startng nodes Ū V and end nodes W V. We say that the set of vertex-dsjont paths s constraned f t s the only set of m vertex-dsjont paths from Ū to W. Next, let U,W V be dsjont subsets of vertces. We say that there exsts a constraned set of m vertex-dsjont paths from U to W f there exsts a constraned set of m vertexdsjont paths n G wth startng nodes Ū U and end nodes W W. In the case that U W, we say that there s a constraned set of m vertex-dsjont paths from U to W f there exsts a constraned set of max{0,m U W } vertexdsjont paths from U \W to W \U. Remark 6. Note that for a set of m vertex-dsjont paths from U to W to be constraned, we do not requre the exstence of a unque set of m vertex-dsjont paths from U to W. In fact, we only requre the exstence of a unque set of vertexdsjont paths between the startng nodesū of the paths and the end nodes W. We wll llustrate the defnton of constraned vertex-dsjont paths n Example. Remark 7. The noton of constraned vertex-dsjont paths s strongly related to the noton of constraned matchngs n bpartte graphs []. In fact, a constraned matchng can be seen as a specal case of a constraned set of vertex-dsjont paths where all paths are of length one. Example. Consder the graph G = (V, E) n Fgure 7. Moreover, consder the subsets of vertces U := {2, } and W := {6,7,8}. Clearly, the paths {(2,),(,6)} and Fg. 7. Graph used n Example. 2 {(,),(,7)} form a set of two vertex-dsjont paths from U to W. In fact, ths set of vertex-dsjont paths s constraned snce there does not exst another set of two vertex-dsjont paths from Ū = {2,} to W = {6,7}. Therefore, we say that there exsts a constraned set of two vertex-dsjont paths from U to W. Note that there are also other sets of vertex-dsjont paths from U to W. For example, also the paths{(2, ),(, 7)} and {(,),(,8)} form a set of two vertex-dsjont paths. However, ths set of vertex-dsjont paths s not constraned. To see ths, note that we have another set of vertex-dsjont paths from Ū = {2,} to W = {7,8}, namely the set consstng of the paths {(2,),(,8)} and {(,),(,7)}