Recognizing Even-Cycle and Even-Cut Matroids

Even-cycle and even-cut matroids are classes of binary matroids that generalize respectively graphic and cographic matroids. We give algorithms to check membership for these classes of matroids. We assume that the matroids are 3-connected and are given by their (0,1)-matrix representations. We first give an algorithm to check membership for p-cographic matroids that is a subclass of even-cut matroids. We use this algorithm to construct algorithms for membership problems for even-cycle and even-cut matroids and the running time of these algorithms is polynomial in the size of the matrix representations. However, we will outline only how theoretical results can be used to develop polynomial time algorithms and omit the details of algorithms

Representations of even-cycle and even-cut matroids

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 $G$. 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

Phase III trial of two versus four additional cycles in patients who are nonprogressive after two cycles of platinum-based chemotherapy in non small-cell lung cancer

PURPOSE: This trial was conducted to determine the optimal duration of chemotherapy in Korean patients with advanced non-small-cell lung cancer (NSCLC). PATIENTS AND METHODS: Patients with stages IIIB to IV NSCLC who had not progressed after two cycles of chemotherapy were randomly assigned to receive either four (arm A) or two (arm B) more cycles of third-generation, platinum-doublet treatment. RESULTS: Of the 452 enrolled patients, 314 were randomly assigned to the groups. One-year survival rates were 59.0% in arm A and 62.4% in arm B, and the difference of 3.4% (95% CI, -8.0 to 4.8) met the predefined criteria for noninferiority. The median time to progression (TTP), however, was 6.2 months (95% CI, 5.7 to 6.7 months) in arm A and 4.6 months (95% CI, 4.4 to 4.8 months) in arm B, the difference of which is statistically significant (P = .001). The frequencies of hematologic and nonhematologic toxicities were similar in the two arms. CONCLUSION: This study confirms the noninferiority of overall survival with four cycles compared with six cycles of chemotherapy for the first-line treatment of advanced NSCLC and supports the current American Society of Clinical Oncology guidelines. Notably, patients receiving six cycles of chemotherapy compared with four cycles showed a favorable TTP, suggesting that further investigation of the new strategies of maintenance therapy with less toxic agents after three to four cycles of induction chemotherapy might be warranted to improve survival, with consideration of both ethnicity and pharmacogenomic signatures