dc.contributor.author Heo, Cheolwon dc.date.accessioned 2021-08-27 18:58:15 (GMT) dc.date.available 2021-08-27 18:58:15 (GMT) dc.date.issued 2021-08-27 dc.date.submitted 2021-08-20 dc.identifier.uri http://hdl.handle.net/10012/17297 dc.description.abstract In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair \$(G,\Sigma)\$ where \$G\$ is a graph and \$\Sigma\$ is a subset of edges of \$G\$. en A cycle \$C\$ of \$G\$ is a subset of edges of \$G\$ such that every vertex of the subgraph of \$G\$ induced by \$C\$ has an even degree. We say that \$C\$ is even in \$(G,\Sigma)\$ if \$|C \cap \Sigma|\$ is even. A matroid \$M\$ is an even-cycle matroid if there exists a signed graph \$(G,\Sigma)\$ such that circuits of \$M\$ precisely corresponds to inclusion-wise minimal non-empty even cycles of \$(G,\Sigma)\$. A graft is a pair \$(G,T)\$ where \$G\$ is a graph and \$T\$ is a subset of vertices of \$G\$ such that each component of \$G\$ contains an even number of vertices in \$T\$. Let \$U\$ be a subset of vertices of \$G\$ and let \$D:= delta_G(U)\$ be a cut of \$G\$. We say that \$D\$ is even in \$(G, T)\$ if \$|U \cap T|\$ is even. A matroid \$M\$ is an even-cut matroid if there exists a graft \$(G,T)\$ such that circuits of \$M\$ corresponds to inclusion-wise minimal non-empty even cuts of \$(G,T)\$.\\ This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations. (a) Isomorphism problem: what is the relationship between two representations? (b) Bounding the number of representations: how many representations can a matroid have? (c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists? These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two \$4\$-connected graphs \$G_1\$ and \$G_2\$ have a pair of signatures \$(\Sigma_1, \Sigma_2)\$ such that \$(G_1, \Sigma_1)\$ and \$(G_2, \Sigma_2)\$ represent the same even-cycle matroids. This also characterize when \$G_1\$ and \$G_2\$ have a pair of terminal sets \$(T_1, T_2)\$ such that \$(G_1,T_1)\$ and \$(G_2,T_2)\$ represent the same even-cut matroid. For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is \$3\$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant \$c\$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by \$c\$. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a \$(0,1)\$-matrix over the finite field \$\gf(2)\$. The time-complexity of these algorithms is polynomial in the size of the input matrix. dc.language.iso en en dc.publisher University of Waterloo en dc.subject even-cycle matroid en dc.subject even-cut matroid en dc.subject recognition algorithm en dc.subject signed graph en dc.subject graft en dc.title Representations of even-cycle and even-cut matroids en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree.discipline Combinatorics and Optimization en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws-etd.embargo.terms 0 en uws.contributor.advisor Guenin, Bertrand uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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