In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.
We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist.
We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.
Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension.
","abstract_html":"<p>In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.</p> <p>We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant <em>h</em> that the number of <em>n</em>-vertex graphs that admit representations with <em>h</em> obstacles is 2<sup>o(n<sup>2</sup>)</sup>. We improve this bound to show that graphs requiring Ω(<em>n</em> / log<sup>2 </sup><em>n</em>) obstacles exist.</p> <p>We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.</p> <p>Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a <em>O</em>(log<em>OPT</em>)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension.</p>","abstract_has_math":false,"creators":["Sarioz, Deniz"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":["Janos Pach"],"committee_chairs":[],"committee_members":["Boris Aronov","Amotz Bar-Noy","Bilal Khan"],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T02:00:35Z","subjects":["Computer Sciences"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/1973","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Janos Pach"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Boris Aronov","Amotz Bar-Noy","Bilal Khan"]},{"key":"dc:creator","label":"Author","values":["Sarioz, Deniz"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2017-03-21T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Graduate School and University Center of The City University of New York"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Computer Sciences"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://academicworks.cuny.edu/gc_etds/1973"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.
We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist.
We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.
Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension.
"]},{"key":"dc:title","label":"Title","values":["Geometric Graph Theory and Wireless Sensor Networks"]}]}],"canonical_facts":{"dc:contributor.advisor":["Janos Pach"],"dc:contributor.committeemember":["Boris Aronov","Amotz Bar-Noy","Bilal Khan"],"dc:creator":["Sarioz, Deniz"],"dc:date.available":["2017-03-21T07:00:00Z"],"dc:description.abstract":["In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.
We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist.
We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.
Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension.
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