On Tradeoffs Between Width and Filllike Graph Parameters
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Abstract
In this work we consider two twocriteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the path width pw(G) and the profile p(G) of the input graph G. We prove that for an arbitrary graph G and an integer t ∈ {1, … , pw(G) + 1}, there exists an interval supergraph G′ of G such that for its clique number it holds \(\omega (G^{\prime })\leq (1+\frac {2}{t})(\textup {\texttt {pw}}({G})+ 1)\) and the number of its edges is bounded by E(G′) ≤ (t + 2)p(G). In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of \(1+\frac {2}{t}\) (plus a small constant) and t + 2, respectively. Note that for a fixed t, both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fillin. We obtain that the treewidth and the fillin problems are also ‘orthogonal’ in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters.
Keywords
Fillin Graph searching Node search Pathwidth Profile Treewidth1 Introduction
Multicriteria optimization problems can be of interest for several reasons, including theoretical insights their study provides or potential practical applications. The selection of the parameters to be simultaneously optimized is dictated by those and can lead to challenging research questions. Our selection is motivated in two ways. First, the choice of the parameters themselves is made according to their importance in graph theory and algorithm design. Second, we paired the parameters according to a potential application that we describe in detail. The first pair of parameters that we minimize is the pathwidth and profile, which can be viewed as computations of linear graph layouts of certain characteristics. The second pair is the treewidth and fillin, which is a treelike graph layout counterpart of the former.
1.1 Related Work
We point out several optimization problems in which pathwidth or treewidth is paired with another parameter or with additional conditions that need to be satisfied. For an example consider a problem of computing a path decomposition with restricted width and length (defined as the number of bags in the path decomposition). It has been first studied in [2] as a problem motivated by an industrial application and called the partner unit problem but finds applications also in scheduling and register allocation [36] or graph searching games [15]. It turns out that there exists a polynomialtime algorithm for computing a path decomposition of width k ≤ 3 and minimum length for an arbitrary input graph, but the problem becomes NPhard for any width k ≥ 4 [15]. If we fix the length to be 2 and ask for a minimum width path (or tree) decomposition, then the problem is also NPhard [24]. Also, a minimum length path decomposition of a width k can be computed efficiently for kconnected graphs [22, 23]. A natural treewidth counterpart of the ‘length’ minimization problem can be seen as minimizing, besides of the width of a tree decomposition, the number of its bags [33]. It is known that such problem is NPcomplete for any fixed k ≥ 4, it is polynomial for k ≤ 2 and for k = 3 it is polynomial for trees and 2connected outerplanar graphs [33]. See [10] for an algorithm of running time 2^{O(n/ log n)} that solves both problems for a fixed k. For a more general approach through using a cost function on tree decompositions see [7].
We refer the reader to a related problem of minimization of width and the diameter of the underlying treestructure of the decomposition [6, 9]. (The diameter of a tree decomposition is defined to be the maximum distance between any two nodes of the three decomposition.)
A very closely related research area includes several graph searching games. We now restrict ourselves to a short and informal introduction and an overview, and in Section 6 we give a formal statement of a graph searching problem, which gives a motivation and a potential application of our results. The problem of graph searching can be informally stated as one in which an agent called the fugitive is moving around the graph with the goal to escape a group of agents called guards or searchers. There are many variations of this problem specifying behavior of the fugitive and the searchers, phase restrictions, speeds of both parties or their other capabilities like visibility, radius of capture etc. Numerous optimization criteria have been studied for these games as well. However, the tradeoffs between different optimization parameters have not yet been throughly analyzed. In this work we refer to one of the two classical formulations of the graph searching problem, namely the node search (see a formal definition in Section 6).
In the original statement of the problem the fugitive is considered invisible (i.e., the searchers can deduce its potential locations only based on the history of their moves) and active, i.e., constantly moving with unbounded speed to counter the searchers’ strategy. It turns out that the minimum number of searchers sufficient to guarantee the capture of the fugitive corresponds to the pathwidth of the underlying graph [12]. Later on, the lazy, also referred to as inert, fugitive variant has been defined in which the fugitive only moves when the searchers are one move apart from catching it. The latter version was first introduced in [12] where the authors show that minimizing the number of searchers precisely corresponds to finding the treewidth of the input graph. Seymour and Thomas proposed in [37] a variant of the game in which the fugitive was visible and active. In the same paper they prove that the visible active variant of the problem is equivalent to the invisible inert variant.
All previously mentioned problems considered the number of searchers used by a strategy to be the optimization criterion. In [16], the authors analyzed the cost defined (informally) as the sum of the guard counts over all steps of the strategy. This graph searching parameter is the one that corresponds to the profile minimization.
Not much is known in terms of twocriteria optimization in the graph searching games. To mention some, examples, there is an analysis of simultaneous minimization of time (number of ‘parallel’ steps) and the number of searchers for the visible variant (this corresponds precisely to the abovementioned width and length minimization of path decompositions) [15] and for the inert one [33] of the node search. For more examples of very closely related twocriteria problems that can be found in the graph searching games see e.g. [8, 14, 30].
The pathwidth or treewidth parameters have been also studied with additional constraints which can be most generally stated as requiring certain connectivity structures to be induced by the bags. These include the parameter of connected pathwidth introduced in [4] in the context of graph searching games and studied further e.g. in [3, 13]. (A path decomposition is connected if the union of the bags from each prefix of the path decomposition induced a connected subgraph.) For a relation with the graph searching games we point out that pathwidth problem is equivalent to the node search game, is equivalent up to an additive difference of 1 to the edge search game and up to a multiplicative factor of 2 (plus a o(1) additive term) to the connected search game, see e.g. [3, 13, 18]. Another example includes the connected treewidth [19].
1.2 Outline
This work mostly deals with simultaneous minimization of widthlike (namely pathwidth and treewidth) and filllike (namely profile and fillin) graph parameters. In order to state our results for pathwidth and profile formally, we introduce the necessary notation in Section 2. For pathwidth and profile we give an upper bound (to be precise, a class of upper bounds that results in a tradeoff between the two parameters) in Section 3 (Theorem 1) and, in Section 4, an example that shows that the two cannot be simultaneously minimized in general (Theorem 2). The latter example also is valid for the tradeoffs between the two corresponding parameters, treewidth and fillin and for this reason we introduce the two in Section 5 and state this result as a corollary (Corollary 1). Section 6 recalls two classical graph searching problems which serve as an example that illustrates a case in which it is natural to optimize the two selected pairs of parameters. These connections are summarized there in Remarks 1 and 2. Thus, this part of the work serves as an additional motivation for this research.
2 Preliminaries
We start with recalling some basic graphtheoretic terms used in this work. For a graph G, we write V (G) and E(G) to denote the sets of its vertices and edges, respectively. We say that a graph G′ is a subgraph of a graph G (and in such case G is a supergraph of G′) if V (G′) ⊆ V (G) and E(G′) ⊆ E(G). Moreover, G′ is a subgraph of Ginduced by X ⊆ V (G) and denoted G[X] (or G′ is an induced subgraph of G for short) if V (G′) = X and E(G′) = {{u, v}∈ E(G)  u, v ∈ X}. A clique is a graph in which any two vertices are adjacent. For a vertex v of a graph G, N_{G}(v) is the set of neighbors of v in G.

◦ \(\bigcup _{i = 1,\ldots ,d}X_{i}=V(G)\),

◦ for each {u, v} ∈ E(G) there exists i ∈ {1, … , d} such that u, v ∈ X_{i},

◦ for each i, j, k, if j is on the path from i to k in \(\mathcal {T}\), then X_{i} ∩ X_{k} ⊆ X_{j}.
2.1 Interval Graphs
A graph G is an interval graph if and only if for each v ∈ V (G) there exists an interval I_{v} = (l_{v}, r_{v}) such that for each edge u, v ∈ V (G) it holds: {u, v}∈ E(G) if and only if I_{u} ∩ I_{v} ≠ ∅. The collection \(\mathcal {I}=\{I_{v}\hspace {0.1cm}\left \right .\hspace {0.1cm} v\in V(G)\}\) is called an interval representation ofG. An interval representation \(\mathcal {I}\) of G is said to be canonical if the endpoints of I_{v} are integers for each v ∈ V (G) and {l_{v}  v ∈ V (G)} = {1, … , n}. This in particular implies that the left endpoints are pairwise different. Denote by \(\mathcal {R}(G)\) the set of all canonical interval representations of G. We will write \(\mathcal {R}(G)\) for a graph G that is not an interval graph to denote the set \(\bigcup _{G^{\prime }\in X}\mathcal {R}(G^{\prime })\), where X is the set of all interval supergraphs of G with the same vertex set as G. If \(\mathcal {I}\) is an interval representation of an interval graph G and v ∈ V (G), then \(\mathcal {I}(v)\) denotes the interval in \(\mathcal {I}\) that corresponds to v. For any interval I, we write left(I) and right(I) to denote its left and right endpoint, respectively. Note that we consider without loss of generality only open intervals in the interval representations.
The next fact follows from [5] and [16].
Proposition 1
 (i)
E(G′) ≤ k, whereG′ is an interval supergraph ofG having the minimum number of edges,
 (ii)
ic(G′) ≤ k,
 (iii)
p(G) ≤ k.
We have the following fact [26, 27, 28, 31, 34]:
Proposition 2
 (i)
iw(G) ≤ k,
 (ii)
pw(G) ≤ k − 1.
2.2 Problem Formulation
For the purposes of this work we need an ‘uniform’ formulation of the two graphtheoretic problems that we study, namely pathwidth and profile, in order to be able to formally apply the two optimization criteria to a single solution to a problem instance. In view of Propositions 1 and 2, we can state the optimization version of our problem as follows:
 Input:

a graph G, integers k and c.
 Question:

does there exist an interval supergraph G′ of G such that iw(G′) ≤ k and E(G′) ≤ c?
3 Pathwidth and Profile Tradeoffs
In this section we prove that for an arbitrary graph G there exists its interval supergraph G′ with width at most \((1+\frac {2}{t})(\textup {\texttt {pw}}({G})+ 1)\) and the number of edges at most (t + 2)p(G) for each t ∈ {1, … , iw(G)}. This is achieved by providing a procedure that finds a desired interval supergraph (the procedure returns an interval representation of this supergraph). Since the goal is to prove an upper bound and not to provide an efficient algorithm, this procedure relies on optimal algorithms for finding a minimum width and minimum cost interval supergraph of a given graph. (The latter problems are NPcomplete, see [20, 25, 32, 39].) Therefore, the running time of this procedure is exponential.

intervals that cover (i, j) entirely or have an empty intersection with it do not change (see Case (i) in Fig. 1),

intervals that contain one of i or j will be extended to cover entire interval, except that we ensure that they have pairwise different left endpoints as required in canonical representations (Cases (ii) and (iii) in Fig. 1),

for the intervals that originally are contained in (i, j), we recompute the interval representation; while doing so we take care of the following: first, the neighborhood relation in the initial graph is respected so that the new interval representation provides an interval supergraph of G, and second, the width of the new interval representation inside (i, j) is minimal (Case (iv) in Fig. 1).
Lemma 1
LetG be any graph and lett ∈ {1, … , iw(G)}. ProcedureICfor the givenG andt returns a canonical interval representation of someinterval supergraph of G.
Proof
We start by noting that the execution of the procedure completes at some point. This follows from an observation that in an iteration of the main loop, a modification to \(\mathcal {I}^{\prime \prime }\) is made by changing the endpoints of some vertex corresponding intervals that are contained in an interval (i, j), where i and j are selected specifically for this iteration. Also, the variable q is set to be j + 1 in this iteration. By the choice of (i, j), and in particular by i ≥ q, we obtain that the subsequent iteration will modify an interval that is to the right of (i, j). This, by an inductive argument, implies that the number of iterations is bounded by the number of endpoints in a canonical representation, which is O(n).
Let \(\mathcal {I}\) be the canonical interval representation of some interval supergraph of G computed at the beginning of Procedure IC. We proceed by induction on the number of iterations of the main loop of Procedure IC, namely, we prove that the interval representation \(\mathcal {I}^{\prime \prime }\) obtained in the sth iteration is a canonical interval representation of some interval supergraph of G. For the purpose of the proof, we use the symbol \(\mathcal {I}_{s}\) to denote the interval representation obtained in the sth iteration, taking \(\mathcal {I}_{0}=\mathcal {I}\).
For the base case of s = 0 we have that \(\mathcal {I}_{0}=\mathcal {I}\) and the claim follows. Hence, let s > 0. Since \(\mathcal {I}_{s}\) consists of V (G) intervals, \(\mathcal {I}_{s}\) is an interval representation of an interval graph G′ on V (G) vertices. We need to prove that G is a subgraph of G′ and that \(\mathcal {I}_{s}\) is canonical.
By the induction hypothesis, there exists an interval supergraph G″ of G and \(\mathcal {I}_{s1}\in \mathcal {R}(G^{\prime \prime })\). Note that V (G) = V (G′) = V (G″).
Theorem 1
LetG be any graph and lett ∈ {1, … , pw(G) + 1} be an integer. There exists an interval supergraphG′ ofG and\(\mathcal {I}^{\prime }\in \mathcal {R}(G^{\prime })\)suchthat\(w(\mathcal {I}^{\prime })\leq (1+\frac {2}{t})(\textup {\texttt {pw}}({G})+ 1)\)and\(ic(\mathcal {I}^{\prime })\leq (t + 2)p({G})\).
Proof
Suppose that Procedure IC is executed for the input G and t. Let \(\mathcal {I}\) be the canonical interval representation of some interval supergraph of G computed at the beginning of Procedure IC. Moreover, take such an \(\mathcal {I}\) that satisfies \(ic(\mathcal {I})=ic(G)\). Let r be the number of iterations performed by the main loop. Let \(\bar {G}_{q}\) and \(\bar {\mathcal {I}}_{q}\), q ∈ {1, … , r}, be the graph \(\bar {G}\) and its interval representation, respectively, computed in the qth iteration of the main loop. Also, let (i_{q}, j_{q}) be the interval used to select \(\bar {G}_{q}\), i.e., \(\bar {G}_{q}\) is the subgraph of G induced by all vertices v such that \(\mathcal {I}^{\prime \prime }(v)\subseteq (i_{q},j_{q})\) for each q ∈ {1, … , r}.
4 Pathwidth and Profile are ‘Orthogonal’
In this section we prove that the two optimization criteria studied in this work cannot be minimized simultaneously for some graphs. In other words, we prove by example, that there exist graphs G such that any interval supergraph G′ of G that has the minimum number of edges (i.e., E(G′) = p(G)) cannot have minimum width (i.e., iw(G) > pw(G) − 1) and vice versa. The example that we construct will be also used in the next section and for this reason we present it here in terms of chordal graphs, which is a class of graphs that generalizes interval graphs. For that we need some additional definitions.
We say that C is an induced cycle of lengthk ≥ 3 in a graph G if C is a subgraph of G and {{u, v}∈ E(G)  u, v ∈ V (C)} = V (C), i.e., the only edges in G between vertices in V (C) are the ones in E(C). A graph is chordal if there is no induced cycle of length greater than 3 in G. Any edge that does not belong to a cycle C and connects two vertices of C is called a chord of C.
We have the following observation.
Lemma 2
IfG′ is a minimal chordal supergraph ofG, then each of the subgraphsG′ [A ∪ C] orG′[B ∪ B′] is either a clique or an union of two disconnected cliques.
Proof
We prove that the subgraph of G′ induced by A ∪ C′ is either a clique or is disconnected and the proof for B ∪ B′ is identical due to the symmetry. If A ∪ C′ induces a clique, then the claim follows so suppose that there exist two vertices a ∈ A ∪ C and c ∈ A ∪ C that are not adjacent in G′. Without loss of generality let a ∈ A and c ∈ C — this is due to the fact that G′[A] and G′[C] are cliques. Take any two vertices b ∈ B and b′∈ B′. Since G′ is chordal, the cycle induced by a, b, b′ and c has a chord in G′. Thus, there is an edge between b and b′ in G′. Since b and b′ are selected arbitrarily, G′[B ∪ B′] is a clique. Note that a supergraph of G that has no edge between any vertex in A and any vertex in C and in which B ∪ B′ induces a clique is chordal. Thus, by the minimality of G′, G′[A ∪ C] consists of two cliques G′[A] and G′[C] with no edges between them, as required.□
Theorem 2
There exists a graphG such that no interval supergraphG′ of G satisfiesiw(G′) = iw(G) andic(G′) = ic(G).
Proof
Consider the graph G = (A ∪ B ∪ B′ ∪ C, E(G)) constructed at the beginning of this section. For any minimal chordal supergraph G′ of G, we say that it is (A, C)connected ((B, B′)connected) if G′[A ∪ C] is a clique (G′[B ∪ B′] is a clique, respectively). Denote by G_{(A, C)} (respectively, \(G_{(B,B^{\prime })}\)) the minimal chordal supergraph of G that is (A, C)connected ((B, B′)connected, respectively) but has no edge joining a vertex in B (respectively A) with a vertex in B′ (respectively C).
Each interval graph is also chordal. On the other hand, both G_{(A, C)} and \(G_{(B,B^{\prime })}\) are interval graphs.
5 Treewidth and Fillin
We refer the reader e.g. to [20, 38] for a definition of the NPcomplete problem of fillin. The treewidth for a given graph G, denoted by tw(G), can be defined as the the minimum k such that there exists a chordal supergraph G′ of G such that the maximum clique ω(G′) of G′ has size at most k + 1. The fillin of G is the minimum m such that there exists a chordal supergraph of G that can be constructed by adding m edges to G. Hence, our corresponding combinatorial problem can be stated as follows:
 Input:

a graph G, integers k and c.
 Question:

does there exist a chordal supergraph G′ of G such that ω(G′) ≤ k and E(G′) ≤ c?
By the same proof as in Theorem 2, we obtain that for some graphs there is no solution to Problem TFM in which k = tw(G) − 1 and c = E(G) + fillin(G).
Corollary 1
There exists a graphG such that no chordal supergraphG′ ofG satisfiestw(G) = ω(G′) − 1 andfillin(G) + E(G) = E(G′).
6 Applications to Graph Searching
6.1 Formal Definitions
 (i)
A_{i} ⊆ V (G) and Z_{i} ⊆ V (G) for each i ∈ {0, … , m},
 (ii)
A_{0} = Z_{0} = ∅, A_{m} = V (G) and Z_{m} = ∅,
 (iii)
(placing/removing searchers) For each i ∈ {1, … , m} there exist v_{i} ∈ V (G) such that {v_{i}} = A_{i} ∖ A_{i− 1}, v_{i} ∈ Z_{i} and Z_{i} ⊆ A_{i− 1} ∪ {v_{i}}.
 (vi)
(possible recontamination) For each i ∈ {1, … , m}, A_{i} consists of v_{i} and each vertex u such that each path connecting u to a vertex in V (G) ∖ A_{i− 1} has an internal vertex in Z_{i}.
 (vi’)
(possible recontamination) For each i ∈ {1, … , m}, A_{i} consists of v_{i} and each vertex u such that each path connecting u to v_{i} has an internal vertex in Z_{i− 1}.
We say that a strategy (active or inert) \(\mathcal {S}=((A_{0},Z_{0}),\ldots ,(A_{m},Z_{m}))\) is monotone if A_{i} ⊆ A_{i+ 1} for each i ∈ {1, … , m − 1}.
6.2 Consequences of our Results
We have the following equivalences:
Theorem 3 (16)
For each graphG, if\(\mathcal {S}\)anactive monotone search strategy of minimum cost,then\(\gamma ({\mathcal {S}})=ic(G)\).
Theorem 4 (26, 27, 28, 34)
For each graphG, if\(\mathcal {S}\)anactive search strategy that uses the minimum number of searchers,then\(\textup {\texttt {ns}}(\mathcal {S})=\textup {\texttt {pw}}({G})+ 1\).
Hence we obtain the following equivalence:
Remark 1
An optimal solution to Problem PPM corresponds to an active search strategy that simultaneously minimizes the number of searchers and the cost.
For the second pair of parameters, we recall the following theorems.
Theorem 5 (17)
For each graphG, if\(\mathcal {S}\)aninert monotone search strategy of minimum cost,then\(\gamma ({\mathcal {S}})=\left {E(G)}\right +\textup {\texttt {fillin}}({G})\).
Theorem 6 (37)
For each graphG, if\(\mathcal {S}\)aninert search strategy that uses the minimum number of searchers,then\(\textup {\texttt {ns}}(\mathcal {S})=\textup {\texttt {tw}}({G})+ 1\).
This leads us to the following theorem:
Remark 2
An optimal solution to Problem TFM corresponds to an inert search strategy that simultaneously minimizes the number of searchers and the cost.
7 Conclusions and Open Problems
We note that the reason why Procedure IC is exponential is its first step, i.e., the computation of the profileminimizing interval representation. Thus, having this as an input, our constructive method is of polynomial running time.
The first open problem we leave is the one of existence of a similar tradeoff between fillin and treewidth to the one we have in Theorem 1. More particularly, is it possible to find chordal supergraphs that approximate both parameters to within constant factors of their optimal values? Our approach used in Procedure IC most likely cannot be extended from interval graphs to chordal graphs as the latter have treelike representations: the constant factor in our tradeoff relies on the fact that we iteratively ‘reorganize’ subintervals of the initial representation. More precisely, each modification performed for an interval (i, j) ‘extends’ the intervals intersecting the points i and j, while in case of chordal graphs we would have to deal with ‘subtrees’ of the corresponding representation. Since such subtrees have potentially many leaves (as opposed to just two represented previously by the endpoints i and j), we cannot ensure keeping constant width of the final representation.
A challenging open problem is the one that refers to the concept of recontamination in the graph searching games that has been posed in [17]: does recontamination help to obtain a minimumcost inert search strategy? Formally, does there exist, for some graph G, an inert search strategy whose cost is smaller than E(G) + fillin(G)? In yet other words, does there exist a graph for which an inert search strategy that minimizes the cost must necessarily allow for recontamination and as a result some vertex v is searched twice in step (iii), i.e., v = v_{i} for two different indices i?
We remark that another example that shows that the problems of finding the minimum fillin and the minimum clique size of an arbitrary graph are ‘orthogonal’ has been independently reported in [11] (see also [29] for some comments).
Notes
References
 1.Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a ktree. SIAM J. Algebraic Discrete Method 8, 277–284 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Aschinger, M., Drescher, C., Gottlob, G., Jeavons, P., Thorstensen, E.: Structural decomposition methods and what they are good for. In: Thomas Schwentick and Christoph Dürr, editors, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011), volume 9 of Leibniz International Proceedings in Informatics (LIPIcs), pp 12–28 (2011)Google Scholar
 3.Barrière, L., Flocchini, P., Fomin, F.V., Fraigniaud, P., Nisse, N., Santoro, N., Thilikos, D.M.: Connected graph searching. Inf. Comput. 219, 1–16 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp 200–209. ACM, New York (2002)Google Scholar
 5.Billionnet, A.: On interval graphs and matrice profiles. RAIRO Rech. Opér. 20(3), 245–256 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Bodlaender, H.L.: NCAlgorithms for Graphs with Small Treewidth. In: WG ’88 14th International Workshop on GraphTheoretic Concepts in Computer Science, Amsterdam, the Netherlands, June 1517, 1988, Proceedings, pp 1–10 (1988)Google Scholar
 7.Bodlaender, H.L., Fomin, F.V.: Tree decompositions with small cost. Discret. Appl. Math. 145(2), 143–154 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 8.Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms 18, 238–255 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Bodlaender, H.L., Hagerup, T.: Tree Decompositions of Small Diameter. In: MFCS’98 23rd International Symposium on Mathematical Foundations of Computer Science 1998, Brno, Czech Republic, August 2428, 1998, Proceedings, pp 702–712 (1998)Google Scholar
 10.Bodlaender, H.L., Nederlof, J.: Subexponential Time Algorithms for Finding Small Tree and Path Decompositions. In: Algorithms  ESA 2015  23Rd Annual European Symposium, Patras, Greece, September 1416, 2015, Proceedings, pp 179–190 (2015)Google Scholar
 11.Bodlaender, H.L., van der Gaag, L., Kloks, T.: Some remarks on minimum edge and minimum clique triangulations. Unpublished resultGoogle Scholar
 12.Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitivesearch games on graphs and related parameters. Theor. Comput. Sci. 172(12), 233–254 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Dereniowski, D.: From pathwidth to connected pathwidth. SIAM J. Discrete Math. 26(4), 1709–1732 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 14.Dereniowski, D., Dyer, D.: On minimum cost edge searching. Theor. Comput. Sci. 495, 37–49 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Dereniowski, D., Kubiak, W., Zwols, Y.: The complexity of minimumlength path decompositions. J. Comput. Syst. Sci. 81(8), 1715–1747 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Fomin, F.V., Golovach, P.A.: Graph searching and interval completion. SIAM J. Discrete Math. 13(4), 454–464 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Fomin, F.V., Heggernes, P., Telle, J.A.: Graph searching, elimination trees, and a generalization of bandwidth. Algorithmica 41(2), 73–87 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor Comput. Sci. 399(3), 236–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 19.Fraigniaud, P., Nisse, N.: Connected Treewidth and Connected Graph Searching. In: Proceedings of the 7Th Latin American Symposium on Theoretical Informatics (LATIN’06), LNCS, volume 3887, pp 479–490. Valdivia, Chile (2006)Google Scholar
 20.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NPcompleteness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
 21.Gibbs, N.E., Poole, W.G. Jr.: Tridiagonalization by permutations. Commun. ACM 17(1), 20–24 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Gupta, A., Nishimura, N.: The complexity of subgraph isomorphism for classes of partial ktrees. Theor. Comput. Sci. 164(1&2), 287–298 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Gupta, A., Nishimura, N., Proskurowski, A., Ragde, P.: Embeddings of kconnected graphs of pathwidth k. Discret. Appl. Math. 145(2), 242–265 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 24.Gustedt, J.: On the pathwidth of chordal graphs. Discrete Appl. Math. 45(3), 233–248 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 25.Kashiwabara, T., Fujisawa, T.: NpCompleteness of the Problem of Finding a MinimumCliqueNumber Interval Graph Containing a Given Graph as a Subgraph. In: Proceedings of the International Conference on Circuits and Systems, pp 657–660 (1979)Google Scholar
 26.Kinnersley, N.G.: The vertex separation number of a graph equals its pathwidth. Inf. Process. Lett. 42(6), 345–350 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
 27.Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and searching. Discrete Math. 55, 181–184 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Kirousis, L.M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47(2), 205–218 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 29.Kloks, T.: Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, Berlin (1994)Google Scholar
 30.Kloks, T., Bodlaender, H.L.: Approximating Treewidth and Pathwidth of Some Classes of Perfect Graphs. In: Proceedings of the Third International Symposium on Algorithms and Computation, ISAAC ’92, pp 116–125. Springer, London (1992)Google Scholar
 31.LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40 (2), 224–245 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 32.Lengauer, T.: Blackwhite pebbles and graph separation. Acta Inf. 16, 465–475 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
 33.Li, B., Moataz, F.Z., Nisse, N., Suchan, K.: Minimum size treedecompositions. Discret. Appl. Math. 245, 109–127 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
 34.Möhring, R.: Graph Problems Related to Gate Matrix Layout and PLA Folding. In: Mayr, E., Noltemeier, H., Syslo, M. (eds.) Computational Graph Theory, Computing Supplementum, vol. 7, pp 17–51 (1990)Google Scholar
 35.Robertson, N., Seymour, P.D.: Graph minors. II. algorithmic aspects of treewidth. J. Algorithm 7(3), 309–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 36.Sethi, R.: Complete register allocation problems. SIAM J. Comput. 4(3), 226–248 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
 37.Seymour, P.D., Thomas, R.: Graph searching and a minmax theorem for treewidth. J. Comb. Theory Ser. B 58(1), 22–33 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 38.Yannakakis, M.: Computing the minimum fillin is npcomplete. SIAM J. Algebraic Discrete Method 2(1), 77–79 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
 39.Yuan, J., Lin, Y., Liu, Y., Wang, S.: Npcompleteness of the profile problem and the fillin problem on cobipartite graphs. J. Math. Study 31(3), 239–243 (1998)MathSciNetzbMATHGoogle Scholar
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