We show that the Temporal Graph Exploration Problem is NP-complete, even when
the underlying graph has pathwidth 2 and at each time step, the current graph
is connected

Bodlaender, Hans L.

van der Zanden, Tom C.

English

arXiv

We show that the Temporal Graph Exploration Problem is NP-complete, even when the underlying graph has pathwidth 2 and at each time step, the current graph is connected

Zanden, Tom C. van der

NARCIS 

On exploring temporal graphs of small pathwidth

Pure OAI Repository

 On exploring temporal graphs of small pathwidthCitation for published version (APA):Bodlaender, H. L., & Zanden, T. C. V. D. (2018). On exploring temporal graphs of small pathwidth. arXiv, 2018,Article 1807.11869. https://doi.org/10.48550/arXiv.1807.11869DOI:10.48550/arXiv.1807.11869Document status and date:Published: 31/07/2018Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023On Exploring Temporal Graphs of SmallPathwidthHans L. Bodlaender∗ Tom C. van der Zanden†AbstractWe show that the Temporal Graph Exploration Problem isNP-complete, even when the underlying graph has pathwidth 2 andat each time step, the current graph is connected.1 IntroductionNetworks can change during time: roads can be blocked or built, friendshipscan wither or new friendships are formed, connections in a computer networkcan go down or be made available, etc. Temporal graphs can serve as a modelfor such changing networks.In this note, we study the complexity of a problem on temporal networks:the Temporal Graph Exploration problem. Recently, Akrida et al. [1]showed that this problem is NP-complete, even when the underlying graphis a star. An important special case, studied by Erlebach et al. [5], is whenat each point in time, the current graph is connected. This case is trivialwhen the underlying graph is a tree; we show that it is already NP-completewhen the underlying graph has pathwidth 2.A temporal graph G is given by a series of graphs G1 = (V,E1), G2 =(V,E2), . . . , GL = (V,EL), each with the same vertex set, but the set ofedges can be different at different time steps. At time step i, only the edgesin Ei exist and can be used. Each i, 1 ≤ i ≤ L is called a time step, Gi is∗Department of Information and Computing Sciences, Utrecht University, P.O.Box80.089, 3508 TB Utrecht, the Netherlands. Email: H.L.Bodlaender@uu.nl, and Depart-ment of Mathematics and Computer Science, Eindhoven University of Technology, Eind-hoven, the Netherlands. The research of this author was partially supported by the Net-works project, supported by the Netherlands Organization for Scientific Research N.W.O.†Department of Information and Computing Sciences, Utrecht University, P.O.Box80.089, 3508 TB Utrecht, the Netherlands. Email: T.C.vanderZanden@uu.nl1arXiv:1807.11869v1  [cs.DS]  31 Jul 2018the current graph at time i. The underlying graph is formed by taking theunion of the graphs at the different time steps. I.e., if we have L time steps,and graphs G1 = (V,E1), G2 = (V,E2), . . . , GL = (V,EL), the underlyinggraph is (V,E1 ∪E2 ∪ · · · ∪EL), so an edge exists in the underlying graph ifit exists in at least one time step. Many graph properties can be studied inthe setting of temporal graphs; this note looks at the problem of exploringthe graph.In temporal graphs, we can define a temporal walk: we have an explorerwho at time step 1 is at a specified vertex s; at each time step i she can moveover an edge in Gi or remain at her current location. In the TemporalGraph Exploration problem, we are given a temporal graph and a start-ing vertex s, and are asked if there exists a temporal walk starting at s thatvisits all vertices within a given time L. A variant is when we require thatthe walk ends at the starting vertex s; we denote this by RTB TemporalGraph Exploration, with RTB the acronym of return to base. (See [1].)Michail and Spirakis [9] introduced the Temporal Graph Explo-ration problem. It is easy to see that even if the graphs do not changeover time, the exploration problem is NP-complete, as it contains Hamil-tonian Path as a special case (set L = n − 1.) Michail and Spirakis [9]showed that the problem does not have a c-approximation, unless P = NP ,and obtained approximation algorithms for several special cases.An important special case is when we require that at each time step,the current graph Gi is connected. Now, if the time L is sufficiently largecompared to the number of vertices n, it is always possible to explore thegraph. Specifically, Erlebach et al. [5] showed that in this case, the graphcan be explored in O(t2n√n log(n)) time steps, where t is the treewidth ofthe underlying graph. Similarly, if the underlying graph is a 2 by n grid thenO(n log3 n) time steps always suffice.Recently, Akrida et al. [1] studied the Temporal Graph Explorationproblem when the underlying graph is a starK1,r. Even when each edge existsin at most six time steps, the problem is NP-complete. We use the followingof their results as starting point.Theorem 1 (Akrida et al. [1]). RTB Temporal Graph Explorationis NP-complete, when the underlying graph is a star, and each edge exists inat most six graphs Gi, and the start and end vertex is the center of the star.For more results, including special cases, approximation algorithms andinapproximability results, see [1, 5, 9], and see [8] for a survey.It is well known that problems that are intractable (e.g., NP-hard) ongeneral graphs become easier (e.g., linear time solvable) when restricted to2graphs of bounded treewidth (see e.g., [3, Chapter 7].) An example is Hamil-tonian Path, which can be solved in O(2O(t)n) time on graphs of treewidtht [2, 4]. Unfortunately, these positive results appear not to carry over to tem-poral graphs: we show that the Temporal Graph Exploration problemis NP-hard, even when the underlying graph has pathwidth 2 (and thus alsotreewidth 2), and at each point in time, the graph is a tree, and thus con-nected.Interestingly, there are other problems on temporal graphs that do be-come tractable when the treewidth is bounded. Specifically, Fluschnik etal. [6] showed that finding a small temporal separator becomes tractable whenthe underlying graph has bounded treewidth; the problem is NP-hard in gen-eral [7].The pathwidth of graphs was defined by Robertson and Seymour [11]. Apath decomposition of a graph G = (V,E) is a sequence of subsets (calledbags) of V (X1, . . . , Xr), such that⋃1≤i≤rXr = V , for all {v, w} ∈ E, thereis an i with v, w ∈ Xi, and if 1 ≤ i1 < i2 < i3 ≤ r, then Xi2 ⊆ Xi1 ∩ Xi3 .The width of a path decomposition (X1, . . . , Xr) equals max1≤i≤r |Xi| − 1;the pathwidth of a graph G is the minimum width of a path decompositionof G. The pathwidth of a graph is an upper bound for its treewidth. (Seee.g. [3, Chapter 7].)K1,r is a star graph with r+ 1 vertices, i.e., we have one vertex of degreer which is adjacent to the remaining r vertices, which have degree 1.2 Hardness resultWe now give our main result.Theorem 2. The Temporal Graph Exploration Problem is NP-complete, even if each graph Gi = (V,Ei) is a tree, and the underlying graphhas pathwidth 2.Proof. We use a reduction from RTB Temporal Graph Explorationfor star graphs. Suppose we have a temporal star K1,n−1, given by a series ofsubgraphs of K1,n−1, G1 = (V,E1), . . ., GL = (V,EL), and a start vertex s,which is the center of the star. We denote the vertices of K1,n by v0, . . . , vn−1,with s = v0.We now build a new temporal graph, as follows. Set Q = L · (n+ 3).The vertex set of the new graph consists of V and Q + 1 new vertices.These will form a path. The new vertices are denoted p0, . . . , pQ and calledpath vertices; the vertices in V are called star vertices.3We now define a temporal graph G ′, given by a series of graphs G′i, 1 ≤i ≤ L′. G′i has the following edges:• For each i, the vertices p0, . . . , pQ form a path: we have edges {pj, pj+1}for 1 ≤ j < Q.• If i ≤ L, all edges in Gi are also edges in G′i.• If i ≤ L, for each star vertex vj ∈ V : if vj is the lowest numberedvertex in a connected component of Gi, we have an edge {vj, pL·(j+2)}.• If i > L, we have an edge from each star vertex vi 6= s to s, and anedge from s to p0.It is not hard to see that each G′i is a tree. If i ≤ L, then G′i is obtainedby adding the path to Gi and one edge from the path to each connectedcomponent of Gi. If i ≥ L, then G′i is obtained taking a path and K1,n andadding an edge between a path and star vertex.2L L L L L 2Lp0 pQsv1 vn−1Figure 1: Illustration to the proof of Theorem 2. Note that edges betweenpath vertices are present at each time step; other edges are present in a subsetof the time steps.The idea behind the proof is that during the first L time steps, we explorethe star vertices as normal, while the path serves to keep the graph connectedbut can not be explored. To explore the path vertices, we must make onesingle pass from p0 to pQ, as we do not have sufficient time to traverse eitherthe section from p0 to p2L−1 or that from pQ−2L+1 twice: traversing theedges between star vertices and path vertices (other than edge {s, p0}) cannotcontribute to a solution.Lemma 3. There is a temporal walk in G ′ that starts at s and visits allvertices in G′ in at most L + Q + 1 time steps, if and only if there is atemporal walk in the temporal star K1,n−1 that starts at s, ends in s andvisits all vertices in K1,n−1 in at most L time steps.4Proof. First, suppose that there is a temporal walk in K1,n−1 that starts ats, ends at s and visits all vertices in at most L time steps. Then, we visitall vertices in G ′, by first making the temporal walk in the star, if necessarywait in s until the end of time step L and at time L + 1 move from s to p1,and then visit all path vertices by traversing the path in the remaining Qtime steps.Suppose we have a temporal walk that starts at s and visits all verticesin G ′ in at most L+Q+ 1 time steps.Claim 4. If we are at a path vertex pi at the end of time step α ≤ L, thenL < i < Q− L.Proof. If we are at a path vertex pi at the end of time step α ≤ L, thenwe moved one or more times from a star vertex to a path vertex during thefirst α time steps. Consider the last of these moves, say that we moved attime step β ≤ α from a star vertex vj to a path vertex pj′ ; between timestep β + 1 and α we stay at path vertices. We have that j′ = L · (j + 2),by construction of the temporal graph. We can make less than L stepsafter reaching pj′ until time step α ≤ L, hence j′ − L < i < j′ + L. Now,L = L · (0 + 2)−L ≤ L · (j+ 2)−L = j′−L < i < j′+L = L · (j+ 2) +L ≤L · (n− 1 + 2) + L = (n+ 2) · L = Q− L.Claim 5. At the end of time step L, we must be in vertex s.Proof. Suppose not. Note that both p0 and pQ are not yet visited, by claim 4.If we are at a star vertex vi 6= s at the end of time step L, then we can onlyvisit the path vertices by first moving to s, then to p0, and then visiting thepath vertices in order; this costs one time step too many. Suppose we areat a path vertex pi at the end of time step L. Suppose we visit p0 beforepQ. Then, we must make at least i steps from pi to p0, and then Q stepsfrom p0 to pQ: in total i+Q > L+Q steps; contradiction. Suppose we visitpQ before p0. Then we must make at least Q − i steps from pi to pQ, andthen Q steps from pQ to p0: in total 2Q − i > L + Q steps; this is again acontradiction.Claim 6. If we move at time step i ≤ L from a star vertex vi to a pathvertex pj, then the first star vertex visited after time step i is again vi, andthis move to vi will be made before the end of time step L.Proof. By Claim 5, we must move to a star vertex before the end of timestep L. If pj′ is a neighbor of a star vertex and j 6= j′, then pj′ is at least Lsteps on the path away from pj, so we cannot reach pj′ before time L, hencewe must move back to the star from pj, and thus move to vi.5Now, we can finish the proof of Lemma 3. Take from the walk in G ′ thefirst L time steps. Change this by replacing each move to a path vertex by astep where the explorer does not move. I.e., when the walk in G ′ moves fromstar vertex vi to a path vertex, then we stay in vi until the time step wherethe walk in G′ moves back to the star — by Claim 6, this is a move to vi. Inthis way, we obtain a walk in K1,n−1 that visits all vertices in L time steps.It remains to show that the underlying graph has pathwidth 2. If we re-move s from the underlying graph, then we obtain a caterpillar: a graph thatcan obtained by taking a path, and adding vertices of degree one, adjacentto a path vertex. These have pathwidth 1 [10]; now add s to all bags and weobtain a path decomposition of the underlying graph of G ′ of width 2.A minor variation of the proof gives also the following result.Theorem 7. The RTB Temporal Graph Exploration Problem isNP-complete, even if each graph Gi = (V,Ei) is a tree, and the underlyinggraph has pathwidth 2.Proof. Modify the proof of Theorem 2 as follows: add one time step; thecurrent graph in the last time step has one edge, from pQ to s.3 ConclusionsIn this note, we showed that the Temporal Graph Exploration Prob-lem is NP-complete, even when we require that at each time step, the graphis connected, or more specifically a tree, and the underlying graph (i.e., thegraph where an edge exists whenever it exists for at least one time step)has pathwidth 2, and hence treewidth 2. This contrasts many other resultsfor graphs of bounded treewidth, including a polynomial time algorithm forfinding small temporal separators for graphs of small treewidth [6].If we require that the graph is connected at each time step, the casethat the treewidth is 1 becomes trivial (as this deletes all temporal effects).Interesting open cases are when the underlying graph is outerplanar, or analmost tree, i.e, can be obtained by adding one edge to a tree.References[1] E. C. Akrida, G. B. Mertzios, and P. G. Spirakis. The temporal explorerwho returns to the base. arXiv, abs/1805.04713, 2018.6[2] H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministicsingle exponential time algorithms for connectivity problems parameter-ized by treewidth. Information and Computation, 243:86 – 111, 2015.[3] M. Cygan, F. V. Fomin,  L. Kowalik, D. Loksthanov, D. Marx,M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms.Springer, 2015.[4] M. Cygan, S. Kratsch, and J. Nederlof. Fast Hamiltonicity checking viabases of perfect matchings. Journal of the ACM, 65(3):12:1–12:46, 2018.[5] T. Erlebach, M. Hoffmann, and F. Kammer. On temporal graph ex-ploration. In Proceedings of the 42nd International Colloquium on Au-tomata, Languages, and Programming, ICALP 2015, Part I, volume9134 of Lecture Notes in Computer Science, pages 444–455. Springer,2015.[6] T. Fluschnik, H. Molter, R. Niedermeier, and P. Zschoche. Tem-poral graph classes: A view through temporal separators. arXiv,abs/1803.00882, 2018. Extended abstract to appear in Proceedings 44thInternational Workshop on Graph-Theoretic Concepts in Computer Sci-ence, WG 2018.[7] D. Kempe, J. Kleinberg, and A. Kumar. Connectivity and inferenceproblems for temporal networks. Journal of Computer and System Sci-ences, 64(4):820–842, 2002.[8] O. Michail. An introduction to temporal graphs: An algorithmic per-spective. Internet Mathematics, 12(4):239–280, 2016.[9] O. Michail and P. G. Spirakis. Traveling salesman problems in temporalgraphs. Theoretical Computer Science, 634:1–23, 2016.[10] A. Proskurowski and J. A. Telle. Classes of graphs with restricted in-terval models. Discrete Mathematics & Theoretical Computer Science,3(4):167–176, 1999.[11] N. Robertson and P. D. Seymour. Graph minors. I. Excluding a forest.Journal of Combinatorial Theory, Series B, 35:39–61, 1983.7

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Algorithms and Complexity

On Exploring Temporal Graphs of Small Pathwidth

Utrecht University Repository

 On exploring temporal graphs of small pathwidthCitation for published version (APA):Bodlaender, H. L., & Zanden, T. C. V. D. (2018). On exploring temporal graphs of small pathwidth. arXiv, 2018,Article 1807.11869. https://doi.org/10.48550/arXiv.1807.11869DOI:10.48550/arXiv.1807.11869Document status and date:Published: 31/07/2018Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023On Exploring Temporal Graphs of SmallPathwidthHans L. Bodlaender∗ Tom C. van der Zanden†AbstractWe show that the Temporal Graph Exploration Problem isNP-complete, even when the underlying graph has pathwidth 2 andat each time step, the current graph is connected.1 IntroductionNetworks can change during time: roads can be blocked or built, friendshipscan wither or new friendships are formed, connections in a computer networkcan go down or be made available, etc. Temporal graphs can serve as a modelfor such changing networks.In this note, we study the complexity of a problem on temporal networks:the Temporal Graph Exploration problem. Recently, Akrida et al. [1]showed that this problem is NP-complete, even when the underlying graphis a star. An important special case, studied by Erlebach et al. [5], is whenat each point in time, the current graph is connected. This case is trivialwhen the underlying graph is a tree; we show that it is already NP-completewhen the underlying graph has pathwidth 2.A temporal graph G is given by a series of graphs G1 = (V,E1), G2 =(V,E2), . . . , GL = (V,EL), each with the same vertex set, but the set ofedges can be different at different time steps. At time step i, only the edgesin Ei exist and can be used. Each i, 1 ≤ i ≤ L is called a time step, Gi is∗Department of Information and Computing Sciences, Utrecht University, P.O.Box80.089, 3508 TB Utrecht, the Netherlands. Email: H.L.Bodlaender@uu.nl, and Depart-ment of Mathematics and Computer Science, Eindhoven University of Technology, Eind-hoven, the Netherlands. The research of this author was partially supported by the Net-works project, supported by the Netherlands Organization for Scientific Research N.W.O.†Department of Information and Computing Sciences, Utrecht University, P.O.Box80.089, 3508 TB Utrecht, the Netherlands. Email: T.C.vanderZanden@uu.nl1arXiv:1807.11869v1  [cs.DS]  31 Jul 2018the current graph at time i. The underlying graph is formed by taking theunion of the graphs at the different time steps. I.e., if we have L time steps,and graphs G1 = (V,E1), G2 = (V,E2), . . . , GL = (V,EL), the underlyinggraph is (V,E1 ∪E2 ∪ · · · ∪EL), so an edge exists in the underlying graph ifit exists in at least one time step. Many graph properties can be studied inthe setting of temporal graphs; this note looks at the problem of exploringthe graph.In temporal graphs, we can define a temporal walk: we have an explorerwho at time step 1 is at a specified vertex s; at each time step i she can moveover an edge in Gi or remain at her current location. In the TemporalGraph Exploration problem, we are given a temporal graph and a start-ing vertex s, and are asked if there exists a temporal walk starting at s thatvisits all vertices within a given time L. A variant is when we require thatthe walk ends at the starting vertex s; we denote this by RTB TemporalGraph Exploration, with RTB the acronym of return to base. (See [1].)Michail and Spirakis [9] introduced the Temporal Graph Explo-ration problem. It is easy to see that even if the graphs do not changeover time, the exploration problem is NP-complete, as it contains Hamil-tonian Path as a special case (set L = n − 1.) Michail and Spirakis [9]showed that the problem does not have a c-approximation, unless P = NP ,and obtained approximation algorithms for several special cases.An important special case is when we require that at each time step,the current graph Gi is connected. Now, if the time L is sufficiently largecompared to the number of vertices n, it is always possible to explore thegraph. Specifically, Erlebach et al. [5] showed that in this case, the graphcan be explored in O(t2n√n log(n)) time steps, where t is the treewidth ofthe underlying graph. Similarly, if the underlying graph is a 2 by n grid thenO(n log3 n) time steps always suffice.Recently, Akrida et al. [1] studied the Temporal Graph Explorationproblem when the underlying graph is a starK1,r. Even when each edge existsin at most six time steps, the problem is NP-complete. We use the followingof their results as starting point.Theorem 1 (Akrida et al. [1]). RTB Temporal Graph Explorationis NP-complete, when the underlying graph is a star, and each edge exists inat most six graphs Gi, and the start and end vertex is the center of the star.For more results, including special cases, approximation algorithms andinapproximability results, see [1, 5, 9], and see [8] for a survey.It is well known that problems that are intractable (e.g., NP-hard) ongeneral graphs become easier (e.g., linear time solvable) when restricted to2graphs of bounded treewidth (see e.g., [3, Chapter 7].) An example is Hamil-tonian Path, which can be solved in O(2O(t)n) time on graphs of treewidtht [2, 4]. Unfortunately, these positive results appear not to carry over to tem-poral graphs: we show that the Temporal Graph Exploration problemis NP-hard, even when the underlying graph has pathwidth 2 (and thus alsotreewidth 2), and at each point in time, the graph is a tree, and thus con-nected.Interestingly, there are other problems on temporal graphs that do be-come tractable when the treewidth is bounded. Specifically, Fluschnik etal. [6] showed that finding a small temporal separator becomes tractable whenthe underlying graph has bounded treewidth; the problem is NP-hard in gen-eral [7].The pathwidth of graphs was defined by Robertson and Seymour [11]. Apath decomposition of a graph G = (V,E) is a sequence of subsets (calledbags) of V (X1, . . . , Xr), such that⋃1≤i≤rXr = V , for all {v, w} ∈ E, thereis an i with v, w ∈ Xi, and if 1 ≤ i1 < i2 < i3 ≤ r, then Xi2 ⊆ Xi1 ∩ Xi3 .The width of a path decomposition (X1, . . . , Xr) equals max1≤i≤r |Xi| − 1;the pathwidth of a graph G is the minimum width of a path decompositionof G. The pathwidth of a graph is an upper bound for its treewidth. (Seee.g. [3, Chapter 7].)K1,r is a star graph with r+ 1 vertices, i.e., we have one vertex of degreer which is adjacent to the remaining r vertices, which have degree 1.2 Hardness resultWe now give our main result.Theorem 2. The Temporal Graph Exploration Problem is NP-complete, even if each graph Gi = (V,Ei) is a tree, and the underlying graphhas pathwidth 2.Proof. We use a reduction from RTB Temporal Graph Explorationfor star graphs. Suppose we have a temporal star K1,n−1, given by a series ofsubgraphs of K1,n−1, G1 = (V,E1), . . ., GL = (V,EL), and a start vertex s,which is the center of the star. We denote the vertices of K1,n by v0, . . . , vn−1,with s = v0.We now build a new temporal graph, as follows. Set Q = L · (n+ 3).The vertex set of the new graph consists of V and Q + 1 new vertices.These will form a path. The new vertices are denoted p0, . . . , pQ and calledpath vertices; the vertices in V are called star vertices.3We now define a temporal graph G ′, given by a series of graphs G′i, 1 ≤i ≤ L′. G′i has the following edges:• For each i, the vertices p0, . . . , pQ form a path: we have edges {pj, pj+1}for 1 ≤ j < Q.• If i ≤ L, all edges in Gi are also edges in G′i.• If i ≤ L, for each star vertex vj ∈ V : if vj is the lowest numberedvertex in a connected component of Gi, we have an edge {vj, pL·(j+2)}.• If i > L, we have an edge from each star vertex vi 6= s to s, and anedge from s to p0.It is not hard to see that each G′i is a tree. If i ≤ L, then G′i is obtainedby adding the path to Gi and one edge from the path to each connectedcomponent of Gi. If i ≥ L, then G′i is obtained taking a path and K1,n andadding an edge between a path and star vertex.2L L L L L 2Lp0 pQsv1 vn−1Figure 1: Illustration to the proof of Theorem 2. Note that edges betweenpath vertices are present at each time step; other edges are present in a subsetof the time steps.The idea behind the proof is that during the first L time steps, we explorethe star vertices as normal, while the path serves to keep the graph connectedbut can not be explored. To explore the path vertices, we must make onesingle pass from p0 to pQ, as we do not have sufficient time to traverse eitherthe section from p0 to p2L−1 or that from pQ−2L+1 twice: traversing theedges between star vertices and path vertices (other than edge {s, p0}) cannotcontribute to a solution.Lemma 3. There is a temporal walk in G ′ that starts at s and visits allvertices in G′ in at most L + Q + 1 time steps, if and only if there is atemporal walk in the temporal star K1,n−1 that starts at s, ends in s andvisits all vertices in K1,n−1 in at most L time steps.4Proof. First, suppose that there is a temporal walk in K1,n−1 that starts ats, ends at s and visits all vertices in at most L time steps. Then, we visitall vertices in G ′, by first making the temporal walk in the star, if necessarywait in s until the end of time step L and at time L + 1 move from s to p1,and then visit all path vertices by traversing the path in the remaining Qtime steps.Suppose we have a temporal walk that starts at s and visits all verticesin G ′ in at most L+Q+ 1 time steps.Claim 4. If we are at a path vertex pi at the end of time step α ≤ L, thenL < i < Q− L.Proof. If we are at a path vertex pi at the end of time step α ≤ L, thenwe moved one or more times from a star vertex to a path vertex during thefirst α time steps. Consider the last of these moves, say that we moved attime step β ≤ α from a star vertex vj to a path vertex pj′ ; between timestep β + 1 and α we stay at path vertices. We have that j′ = L · (j + 2),by construction of the temporal graph. We can make less than L stepsafter reaching pj′ until time step α ≤ L, hence j′ − L < i < j′ + L. Now,L = L · (0 + 2)−L ≤ L · (j+ 2)−L = j′−L < i < j′+L = L · (j+ 2) +L ≤L · (n− 1 + 2) + L = (n+ 2) · L = Q− L.Claim 5. At the end of time step L, we must be in vertex s.Proof. Suppose not. Note that both p0 and pQ are not yet visited, by claim 4.If we are at a star vertex vi 6= s at the end of time step L, then we can onlyvisit the path vertices by first moving to s, then to p0, and then visiting thepath vertices in order; this costs one time step too many. Suppose we areat a path vertex pi at the end of time step L. Suppose we visit p0 beforepQ. Then, we must make at least i steps from pi to p0, and then Q stepsfrom p0 to pQ: in total i+Q > L+Q steps; contradiction. Suppose we visitpQ before p0. Then we must make at least Q − i steps from pi to pQ, andthen Q steps from pQ to p0: in total 2Q − i > L + Q steps; this is again acontradiction.Claim 6. If we move at time step i ≤ L from a star vertex vi to a pathvertex pj, then the first star vertex visited after time step i is again vi, andthis move to vi will be made before the end of time step L.Proof. By Claim 5, we must move to a star vertex before the end of timestep L. If pj′ is a neighbor of a star vertex and j 6= j′, then pj′ is at least Lsteps on the path away from pj, so we cannot reach pj′ before time L, hencewe must move back to the star from pj, and thus move to vi.5Now, we can finish the proof of Lemma 3. Take from the walk in G ′ thefirst L time steps. Change this by replacing each move to a path vertex by astep where the explorer does not move. I.e., when the walk in G ′ moves fromstar vertex vi to a path vertex, then we stay in vi until the time step wherethe walk in G′ moves back to the star — by Claim 6, this is a move to vi. Inthis way, we obtain a walk in K1,n−1 that visits all vertices in L time steps.It remains to show that the underlying graph has pathwidth 2. If we re-move s from the underlying graph, then we obtain a caterpillar: a graph thatcan obtained by taking a path, and adding vertices of degree one, adjacentto a path vertex. These have pathwidth 1 [10]; now add s to all bags and weobtain a path decomposition of the underlying graph of G ′ of width 2.A minor variation of the proof gives also the following result.Theorem 7. The RTB Temporal Graph Exploration Problem isNP-complete, even if each graph Gi = (V,Ei) is a tree, and the underlyinggraph has pathwidth 2.Proof. Modify the proof of Theorem 2 as follows: add one time step; thecurrent graph in the last time step has one edge, from pQ to s.3 ConclusionsIn this note, we showed that the Temporal Graph Exploration Prob-lem is NP-complete, even when we require that at each time step, the graphis connected, or more specifically a tree, and the underlying graph (i.e., thegraph where an edge exists whenever it exists for at least one time step)has pathwidth 2, and hence treewidth 2. This contrasts many other resultsfor graphs of bounded treewidth, including a polynomial time algorithm forfinding small temporal separators for graphs of small treewidth [6].If we require that the graph is connected at each time step, the casethat the treewidth is 1 becomes trivial (as this deletes all temporal effects).Interesting open cases are when the underlying graph is outerplanar, or analmost tree, i.e, can be obtained by adding one edge to a tree.References[1] E. C. Akrida, G. B. Mertzios, and P. G. Spirakis. The temporal explorerwho returns to the base. arXiv, abs/1805.04713, 2018.6[2] H. L. Bodlaender, M. Cygan, S. Kratsch, and J. Nederlof. Deterministicsingle exponential time algorithms for connectivity problems parameter-ized by treewidth. Information and Computation, 243:86 – 111, 2015.[3] M. Cygan, F. V. Fomin,  L. Kowalik, D. Loksthanov, D. Marx,M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms.Springer, 2015.[4] M. Cygan, S. Kratsch, and J. Nederlof. Fast Hamiltonicity checking viabases of perfect matchings. Journal of the ACM, 65(3):12:1–12:46, 2018.[5] T. Erlebach, M. Hoffmann, and F. Kammer. On temporal graph ex-ploration. In Proceedings of the 42nd International Colloquium on Au-tomata, Languages, and Programming, ICALP 2015, Part I, volume9134 of Lecture Notes in Computer Science, pages 444–455. Springer,2015.[6] T. Fluschnik, H. Molter, R. Niedermeier, and P. Zschoche. Tem-poral graph classes: A view through temporal separators. arXiv,abs/1803.00882, 2018. Extended abstract to appear in Proceedings 44thInternational Workshop on Graph-Theoretic Concepts in Computer Sci-ence, WG 2018.[7] D. Kempe, J. Kleinberg, and A. Kumar. Connectivity and inferenceproblems for temporal networks. Journal of Computer and System Sci-ences, 64(4):820–842, 2002.[8] O. Michail. An introduction to temporal graphs: An algorithmic per-spective. Internet Mathematics, 12(4):239–280, 2016.[9] O. Michail and P. G. Spirakis. Traveling salesman problems in temporalgraphs. Theoretical Computer Science, 634:1–23, 2016.[10] A. Proskurowski and J. A. Telle. Classes of graphs with restricted in-terval models. Discrete Mathematics & Theoretical Computer Science,3(4):167–176, 1999.[11] N. Robertson and P. D. Seymour. Graph minors. I. Excluding a forest.Journal of Combinatorial Theory, Series B, 35:39–61, 1983.7

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