Homothetic polygons and beyond: Maximal cliques in intersection graphs
We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in inter...
Ausführliche Beschreibung
Autor*in: |
Brimkov, Valentin E. [verfasserIn] |
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E-Artikel |
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Englisch |
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2018transfer abstract |
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15 |
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Übergeordnetes Werk: |
Enthalten in: Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures - Miguel, F.L. ELSEVIER, 2013transfer abstract, [S.l.] |
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Übergeordnetes Werk: |
volume:247 ; year:2018 ; day:1 ; month:10 ; pages:263-277 ; extent:15 |
Links: |
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DOI / URN: |
10.1016/j.dam.2018.03.046 |
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ELV043761747 |
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245 | 1 | 0 | |a Homothetic polygons and beyond: Maximal cliques in intersection graphs |
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520 | |a We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. | ||
520 | |a We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. | ||
650 | 7 | |a Maximal clique |2 Elsevier | |
650 | 7 | |a P h o m graphs |2 Elsevier | |
650 | 7 | |a Geometric intersection graphs |2 Elsevier | |
700 | 1 | |a Junosza-Szaniawski, Konstanty |4 oth | |
700 | 1 | |a Kafer, Sean |4 oth | |
700 | 1 | |a Kratochvíl, Jan |4 oth | |
700 | 1 | |a Pergel, Martin |4 oth | |
700 | 1 | |a Rzążewski, Paweł |4 oth | |
700 | 1 | |a Szczepankiewicz, Matthew |4 oth | |
700 | 1 | |a Terhaar, Joshua |4 oth | |
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10.1016/j.dam.2018.03.046 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001144.pica (DE-627)ELV043761747 (ELSEVIER)S0166-218X(18)30145-8 DE-627 ger DE-627 rakwb eng 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Brimkov, Valentin E. verfasserin aut Homothetic polygons and beyond: Maximal cliques in intersection graphs 2018transfer abstract 15 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. Maximal clique Elsevier P h o m graphs Elsevier Geometric intersection graphs Elsevier Junosza-Szaniawski, Konstanty oth Kafer, Sean oth Kratochvíl, Jan oth Pergel, Martin oth Rzążewski, Paweł oth Szczepankiewicz, Matthew oth Terhaar, Joshua oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:247 year:2018 day:1 month:10 pages:263-277 extent:15 https://doi.org/10.1016/j.dam.2018.03.046 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 247 2018 1 1001 263-277 15 |
spelling |
10.1016/j.dam.2018.03.046 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001144.pica (DE-627)ELV043761747 (ELSEVIER)S0166-218X(18)30145-8 DE-627 ger DE-627 rakwb eng 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Brimkov, Valentin E. verfasserin aut Homothetic polygons and beyond: Maximal cliques in intersection graphs 2018transfer abstract 15 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. Maximal clique Elsevier P h o m graphs Elsevier Geometric intersection graphs Elsevier Junosza-Szaniawski, Konstanty oth Kafer, Sean oth Kratochvíl, Jan oth Pergel, Martin oth Rzążewski, Paweł oth Szczepankiewicz, Matthew oth Terhaar, Joshua oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:247 year:2018 day:1 month:10 pages:263-277 extent:15 https://doi.org/10.1016/j.dam.2018.03.046 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 247 2018 1 1001 263-277 15 |
allfields_unstemmed |
10.1016/j.dam.2018.03.046 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001144.pica (DE-627)ELV043761747 (ELSEVIER)S0166-218X(18)30145-8 DE-627 ger DE-627 rakwb eng 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Brimkov, Valentin E. verfasserin aut Homothetic polygons and beyond: Maximal cliques in intersection graphs 2018transfer abstract 15 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. Maximal clique Elsevier P h o m graphs Elsevier Geometric intersection graphs Elsevier Junosza-Szaniawski, Konstanty oth Kafer, Sean oth Kratochvíl, Jan oth Pergel, Martin oth Rzążewski, Paweł oth Szczepankiewicz, Matthew oth Terhaar, Joshua oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:247 year:2018 day:1 month:10 pages:263-277 extent:15 https://doi.org/10.1016/j.dam.2018.03.046 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 247 2018 1 1001 263-277 15 |
allfieldsGer |
10.1016/j.dam.2018.03.046 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001144.pica (DE-627)ELV043761747 (ELSEVIER)S0166-218X(18)30145-8 DE-627 ger DE-627 rakwb eng 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Brimkov, Valentin E. verfasserin aut Homothetic polygons and beyond: Maximal cliques in intersection graphs 2018transfer abstract 15 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. Maximal clique Elsevier P h o m graphs Elsevier Geometric intersection graphs Elsevier Junosza-Szaniawski, Konstanty oth Kafer, Sean oth Kratochvíl, Jan oth Pergel, Martin oth Rzążewski, Paweł oth Szczepankiewicz, Matthew oth Terhaar, Joshua oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:247 year:2018 day:1 month:10 pages:263-277 extent:15 https://doi.org/10.1016/j.dam.2018.03.046 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 247 2018 1 1001 263-277 15 |
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10.1016/j.dam.2018.03.046 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001144.pica (DE-627)ELV043761747 (ELSEVIER)S0166-218X(18)30145-8 DE-627 ger DE-627 rakwb eng 070 VZ 660 VZ 333.7 610 VZ 43.12 bkl 43.13 bkl 44.13 bkl Brimkov, Valentin E. verfasserin aut Homothetic polygons and beyond: Maximal cliques in intersection graphs 2018transfer abstract 15 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. Maximal clique Elsevier P h o m graphs Elsevier Geometric intersection graphs Elsevier Junosza-Szaniawski, Konstanty oth Kafer, Sean oth Kratochvíl, Jan oth Pergel, Martin oth Rzążewski, Paweł oth Szczepankiewicz, Matthew oth Terhaar, Joshua oth Enthalten in Elsevier Miguel, F.L. ELSEVIER Electroless deposition of a Ag matrix on semiconducting one-dimensional nanostructures 2013transfer abstract [S.l.] (DE-627)ELV011300345 volume:247 year:2018 day:1 month:10 pages:263-277 extent:15 https://doi.org/10.1016/j.dam.2018.03.046 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-GGO GBV_ILN_20 GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_130 43.12 Umweltchemie VZ 43.13 Umwelttoxikologie VZ 44.13 Medizinische Ökologie VZ AR 247 2018 1 1001 263-277 15 |
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We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. |
abstractGer |
We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. |
abstract_unstemmed |
We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n -vertex graph which is an intersection graph of homothetic copies of P contains at most n k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called k DIR - CONV , which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques. |
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Homothetic polygons and beyond: Maximal cliques in intersection graphs |
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