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Sphere-Cut Decompositions and Dominating Sets in Planar Graphs

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Sphere-cut Decompositions and Dominating Sets in Planar Graphs Michalis Samaris R.N. 201314 Scientific committee: Dimitrios M. Thilikos, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Supervisor: Stavros G. Kolliopoulos, Dimitrios M. Thilikos, Associate Professor, Professor, Dep. of Informatics and Dep. of Mathematics, National and Telecommunications, National and Kapodistrian University of Athens. Kapodistrian University of Athens. white Lefteris M. Kirousis, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Aposunjèseic sfairik¸n tom¸n kai σύνοla kuriarqÐac se epÐpeda γραφήματa Miχάλης Σάμαρης A.M. 201314 Τριμελής Epiτροπή: Δημήτρioc M. Jhlυκός, Epiblèpwn: Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. Δημήτρioc M. Jhlυκός, Staύρoc G. Kolliόποuloc, Kajhγητής tou Τμήμatoc Anaπληρωτής Kajhγητής, Tm. Plhroforiκής Majhmatik¸n tou PanepisthmÐou kai Thl/ni¸n, E.K.P.A. Ajhn¸n Leutèrhc M. Kuroύσης, white Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. PerÐlhyh 'Ena σημαντικό apotèlesma sth JewrÐa Γραφημάτwn apoteleÐ h apόdeixh thc eikasÐac tou Wagner από touc Neil Robertson kai Paul D. Seymour. sth σειρά ergasi¸n ‘Ελλάσσοna Γραφήματα’ apo to 1983 e¸c to 2011. H eikasÐa αυτή lèei όti sthn κλάση twn γραφημάtwn den υπάρχει άπειρη antialusÐda ¸c proc th sqèsh twn ελλασόnwn γραφημάτwn. H JewrÐa pou αναπτύχθηκε gia thn απόδειξη αυτής thc eikasÐac eÐqe kai èqei ακόμα σημαντικό antÐktupo tόσο sthn δομική όσο kai sthn algoriθμική JewrÐa Γραφημάτwn, άλλα kai se άλλα pedÐa όπως h Παραμετρική Poλυπλοκόthta. Sta πλάιsia thc απόδειξης oi suggrafeÐc eiσήγαγαν kai nèec paramètrouc πλά- touc. Se autèc ήτan h κλαδοαποσύνθεση kai to κλαδοπλάτoc ενός γραφήματoc. H παράμετρος αυτή χρησιμοποιήθηκε idiaÐtera sto σχεδιασμό algorÐjmwn kai sthn χρήση thc τεχνικής ‘διαίρει kai basÐleue’. Επιπλεόn eiσήχθησαν nèec paremfereÐc ènnoiec όπως oi aposunjèseic sfairik¸n tom¸n pou eÐnai kladoaposunjèseic sthn klash twn epÐpedwn γραφημάτwn pou èqoun κάποιες epiplèon ιδιόthtec. Sthn exèlixh thc èreunac υπήρξαν σημαντικά apotelèsma σχετικά me to klado- πλάτoc sthn κλάση twn epÐpedwn γραφημάτwn. Oi Fedor V. Fomin kai Δημήτριoc M. Θηλυκός apèdeixan όti to κλαδοπλάτoc ενός epÐpedou γραφήματoc me n korufèc eÐnai to poλύ p4:5 n. Βασιζόμενος se autό to apotèlesma, o Dhμήτριoc M. Jh- · λυκός susqètisqe to κλαδοπλάτoc me mia άλλη παράμετρο se epÐpeda γραφήματa, to r-aktiνικό σύνοlo kuriarqÐac. Apèdeixe όti an èna embaptismèno epÐpedo γράφημα èqei r- aktiνικό σύνοlo kuriarqÐac to poλύ k, tόte to kladοπλάtoc tou γραφήματoc ja eÐnai to poλύ r p4:5 k. · · H παρούσα διπλωματική ergasÐa κάνει mia poioτική epèktash tou apotelèsmatoc autoύ. Αποδεικνύουμε όti to παραπάνω όριo mporeÐ na anazhthjeÐ se èna δάσοc pou eÐnai upoγράφημα tou dèntrou miac αποσύνθεσης sfairik¸n tom¸n tou γραφήματoc, όπου to mègejoc tou eÐnai γραμμικό wc proc to k. i Abstract An important result in Graph Theory is the proof of Wagner's Conjecture by Neil Robertson and Paul D. Seymour in Graph Minor Series from 1983 until 2011. This conjecture state that there is no infinite anti-chain in the class of graphs under the minor relation. The theory that was built for the proof of this conjecture had, and continues to have, an important impact not only in structural and algorithmic Graph Theory, but also in other fields such as Parameterized Complexity. In the context of this proof, the authors have introduced some new width parameters. Within these were branchwidth and branch decompositions. This parameter was used for algorithm design via the \divide and conquer" technique. Moreover, the authors have introduced, similar to branch decompositions, concepts such as sphere-cut decompositions which are a special type of branch decomposi- tions in planar graphs that have some additional properties. In the course of the research there was a lot of important results about branch- width in the class of planar graphs. Fedor V. Fomin and Dimitrios M. Thilikos proved that the branchwidth of a n-vertex planar graph is at most p4:5 n. Based · on this result Dimitrios M. Thilikos connected the branchwidth with r-radial dom- inating set which is another parameter in plane graphs. He proved that if a plane graph has an r-radial dominating set of size at most k, then the branchwidth of the graph is at most r p4:5 k. · · The purpose of this thesis is to provide a qualitative extension of this result. What we show is that this upper bound is attained by a number of edges of a sphere-cut decomposition, that is a linear function of k. iii Πρόlogoc H παρούσα διπλωματική ergasÐa ekpoνήθηκε sta plaÐsia thc oλοκλήρωση twn spoud¸n mou sto Metaptuχιακό Prόgramma Logikής, AlgorÐjmwn kai JewrÐa Upologismoύ (M.P.L.A.). Euqarist¸ idiaÐtera ton epiblèponta thc παρούσας ergasÐac, Καθηγητή k. Δημή- trio M. Θηλυκό, gia thn στήριξη pou mou prosèfere, gia to gegoνός oti me eiσήγαγε sto ιδιόμορφο pedÐo thc èreunac αλλά kai gia tic endiafèrousec antiparajèseic se mh majhmatikèc suzhtήσεic. Euqarist¸ ton Καθηγητή k. Leutèrh M. Κυρούση, gia th summetoχή tou sthn τριμελή epiτροπή, gegoνός pou apoteleÐ τιμή gia emèna, thn stήrixh tou kai to ei- likrinèc endiafèron tou gia thn foÐthsh mou sto par¸n metαπτυχιακό pρόgramma. Euqarist¸ ton Αναπληρωτή Καθηγητή k. Stαύρο G. Koλλιόπουλο pou me τιμά me thn summetoχή tou sthn τριμελή epiτροπή αλλά kai gia ton εξαιρετικά endiafèron τρόπο didaskalÐac tou. Euqarist¸ epÐshc ton Δημήτρη Z¸ro gia thn διάθεση tou na βοηθήσει se όti epimèrouc ζήτημα mporeÐ na proèkupte katά thn διάρκεια thc foithshc mou. Ja μπορούσα na γράψω poλλά λόgia gia poλλούς ακόμα anjr¸pouc, επειδή όμως ta praγματικά euqarist¸ eÐnai πράξειc kai όqi oi lÐgec grammèc ενός proλόgou ja proτιμήσω na eÐmai συνοπτικός. Jewr¸ όμως anagkaÐec kai tic παρακάτw anaforèc. Katά th διάρκεια thc foÐthshc mou υπήρξε to ζήτημα thc ananèwshc tou metap- τυχιακού proγράμματoc tou M.P.L.A. pou τελικά den katèsth efiktό. Gia thn ag- wnÐa pou μοιραστήκαμε, kai tic aneparkeÐc, όπως apodeÐqjeikan, prospάθειες na apotrèyoume to gegoνός autό, ja ήθελα na euqarisτήσω touc sumfoithtèc mou. Tèloc den mpor¸ na paraleÐyw na euqariστήσω touc δικούς mou anjr¸pouc gia thn αδιαπραγμάτευτη stήριxh twn epilog¸n mou. Miχάλης Σάμαρης Αθήνα, 14 IounÐou 2016 v Contents 1 Introduction 1 1.1 About graphs . .1 1.2 Main description of our result . .3 1.3 Structure of the thesis . .8 2 Basic definitions and results 11 2.1 Basic definitions . 11 2.2 Basic concepts . 14 2.3 Duality . 16 3 Partially Ordered Sets 19 4 Width-parameters and Decompositions 23 4.1 Branchwidth ............................. 25 4.2 Sphere-cut decompositions. .................... 27 5 Radial Dominating Set 29 6 Weight and Capacity 33 6.1 q-weight . 33 6.2 Before the main proof . 33 7 Main Result 41 7.1 Weakly dominated and strong dominated triconnected components 41 7.2 Main Proof . 42 7.3 Conclusion and further work . 44 vii List of Figures 1.1 The drawing riddle . .1 1.2 A graph G and a sphere-cut decomposition of it. Given that Nei is the sphere-cut obtained by the edge ei of T , we denote by Vi the set of vertices that it meets and ci the cost of the corresponding sphere-cut. Therefore, we have that V = v ; v ; c = 2;V = 1 f 1 3g 1 2 v ; v ; c = 2;V = v ; v ; v ; c = 3;V = v ; v ; c = 2;V = f 1 2g 2 3 f 1 2 3g 3 4 f 2 3g 4 5 v ; v ; v ; c = 3;V = v ; v ; v ; c = 3;V = v ; v ; c = f 1 2 3g 5 6 f 1 3 4g 6 7 f 3 3g 7 2;V = v ; v ; c = 2;V = v ; v ; v ; c = 3;V = v ; v ; v ; c = 8 f 1 4g 8 9 f 2 3 4g 9 10 f 2 3 5g 10 3;V = v ; v ; c = 2;V = v ; v ; c = 2;V = v ; v ; c = 11 f 3 5g 11 12 f 2 5g 12 13 f 4 5g 13 2. The cost of (T; τ) is 3. .5 1.3 Ne9 is drawn with red. .6 1.4 The radial distance between x and y is 4. One of the respective sequences is x; f1; z; f2; y........................6 1.5 The vertices x and y are a 3-radial dominating set in this graph. .7 1.6 The edges with > q correspond to sphere-cuts with cost > q. The edges that have been drawn fat are the edges of the q-core that we have choose. This q-core has 3 connected components with weight 10,4,7 respectively. The weight of the q-core is 21. .8 2.1 Examples of graphs that are used frequently. 12 2.2 A multigraph with one loop and three multiple edges. 14 2.3 A graph and its triconnected components. 15 2.4 An example of a triangulated graph. 16 2.5 A graph drawn with black and its dual with red. 17 2.6 The radial graph of graphs in Figure 2.5. The black vertices are the vertices of G. The red verticesare the vertices of G∗. Observe that all the edges of RG have one black and one red endpoint. 17 ix 3.1 The graph in (d) is minor of the graph in (a) that is obtained by deleting the blue vertices and edges in (b) and contracting the red edges in (c). 20 3.2 The obstruction set of outerplanar graphs, K4 and K2;3....... 21 4.1 A graph G with a tree decomposition of it. As easily can be seen from the decomposition, every bag has at most 3 vertices, so tw(G)=2............................... 24 4.2 A graph G and a branch decomposition of it. 26 6.1 D and D0 are branch decompositions of G2 and G1 respectively, where G = G e.
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  • Trees and Graphs

    Trees and Graphs

    Module 8: Trees and Graphs Theme 1: Basic Properties of Trees A (rooted) tree is a finite set of nodes such that ¯ there is a specially designated node called the root. ¯ d Ì ;Ì ;::: ;Ì ½ ¾ the remaining nodes are partitioned into disjoint sets d such that each of these Ì ;Ì ;::: ;Ì d ½ ¾ sets is a tree. The sets d are called subtrees,and the degree of the root. The above is an example of a recursive definition, as we have already seen in previous modules. A Ì Ì Ì B C ¾ ¿ Example 1: In Figure 1 we show a tree rooted at with three subtrees ½ , and rooted at , and D , respectively. We now introduce some terminology for trees: ¯ A tree consists of nodes or vertices that store information and often are labeled by a number or a letter. In Figure 1 the nodes are labeled as A;B;:::;Å. ¯ An edge is an unordered pair of nodes (usually denoted as a segment connecting two nodes). A; B µ For example, ´ is an edge in Figure 1. A ¯ The number of subtrees of a node is called its degree. For example, node is of degree three, while node E is of degree two. The maximum degree of all nodes is called the degree of the tree. Ã ; Ä; F ; G; Å ; Á Â ¯ A leaf or a terminal node is a node of degree zero. Nodes and are leaves in Figure 1. B ¯ A node that is not a leaf is called an interior node or an internal node (e.g., see nodes and D ).
  • Separator Theorems and Turán-Type Results for Planar Intersection Graphs

    Separator Theorems and Turán-Type Results for Planar Intersection Graphs

    SEPARATOR THEOREMS AND TURAN-TYPE¶ RESULTS FOR PLANAR INTERSECTION GRAPHS JACOB FOX AND JANOS PACH Abstract. We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m p 2 crossings has a partition into three parts C = S [ C1 [ C2 such that jSj = O( m); maxfjC1j; jC2jg · 3 jCj; and no element of C1 has a point in common with any element of C2. These results are used to obtain various properties of intersection patterns of geometric objects in the plane. In particular, we prove that if a graph G can be obtained as the intersection graph of n convex sets in the plane and it contains no complete bipartite graph Kt;t as a subgraph, then the number of edges of G cannot exceed ctn, for a suitable constant ct. 1. Introduction Given a collection C = fγ1; : : : ; γng of compact simply connected sets in the plane, their intersection graph G = G(C) is a graph on the vertex set C, where γi and γj (i 6= j) are connected by an edge if and only if γi \ γj 6= ;. For any graph H, a graph G is called H-free if it does not have a subgraph isomorphic to H. Pach and Sharir [13] started investigating the maximum number of edges an H-free intersection graph G(C) on n vertices can have. If H is not bipartite, then the assumption that G is an intersection graph of compact convex sets in the plane does not signi¯cantly e®ect the answer.