A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4.
","abstract_html":"<p>A decomposition <em>D</em> of a graph <em>H</em> by a graph <em>G</em> is a partition of the edge set of <em>H</em> such that the subgraph induced by the edges in each part of the partition is isomorphic to <em>G</em>. A <em>mixed graph</em> on <em>V</em> vertices is an ordered pair (<em>V</em>,<em>C</em>), where <em>V</em> is a set of vertices, |<em>V</em>| = <em>v</em>, and <em>C</em> is a set of ordered and unordered pairs, denoted (<em>x</em>, <em>y</em>) and [<em>x</em>, <em>y</em>] respectively, of elements of <em>V</em> [8]. An ordered pair (<em>x</em>, <em>y</em>) ∈ <em>C</em> is called an <em>arc</em> of (<em>V</em>,<em>C</em>) and an unordered pair [<em>x</em>, <em>y</em>] ∈ <em>C</em> is called an <em>edge</em> of graph (<em>V</em>,<em>C</em>). A path on <em>n</em> vertices is denoted as <em>P<sub>n</sub></em>. A <em>partial orientation</em> on <em>G</em> is obtained by replacing each edge [<em>x</em>, <em>y</em>] ∈ <em>E</em>(<em>G</em>) with either (<em>x</em>, <em>y</em>), (<em>y</em>, <em>x</em>), or [<em>x</em>, <em>y</em>] in such a way that there are twice as many arcs as edges. The <em>complete mixed graph</em> on <em>v</em> vertices, denoted <em>M<sub>v</sub></em>, is the mixed graph (<em>V</em>,<em>C</em>) where for every pair of distinct vertices <em>v</em><sub>1</sub>, <em>v</em><sub>2</sub> ∈ <em>V</em> , we have {(<em>v</em><sub>1</sub>, <em>v</em><sub>2</sub>), (<em>v</em><sub>2</sub>, <em>v</em><sub>1</sub>), [<em>v</em><sub>1</sub>, <em>v</em><sub>2</sub>]} ⊂ <em>C</em>. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of <em>M<sub>v</sub></em> by all possible partial orientations of <em>P</em><sub>4</sub>.</p>","abstract_has_math":false,"creators":["Meadows, Adam M."],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - unrestricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-05-09T07:00:00Z","date_published":"2009-05-09T07:00:00Z","updated_at":"2026-07-24T02:21:03Z","subjects":["graph theory","graph decomposition","graceful labeling","mixed graphs","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/1870","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Meadows, Adam M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2009-05-09T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - unrestricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["graph theory","graph decomposition","graceful labeling","mixed graphs","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/3222/viewcontent/MeadowsA041309f.PDF","https://dc.etsu.edu/etd/1870"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4.
"]},{"key":"dc:title","label":"Title","values":["Decompositions of Mixed Graphs with Partial Orientations of the P4."]}]}],"canonical_facts":{"dc:creator":["Meadows, Adam M."],"dc:date.issued":["2009-05-09T07:00:00Z"],"dc:description.abstract":["A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/3222/viewcontent/MeadowsA041309f.PDF","https://dc.etsu.edu/etd/1870"],"dc:rights":["Copyright by the authors."],"dc:subject":["graph theory","graph decomposition","graceful labeling","mixed graphs","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Decompositions of Mixed Graphs with Partial Orientations of the P4."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:21:03Z"}