Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality

The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs, and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In this paper, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs, using a recently introduced tool in matching theory, the basilica decomposition. Our result enables us to understand all known canonical decompositions in a unified way. Furthermore, we apply our result to derive a new theorem regarding barriers. The duality theorem for the maximum matching problem is the celebrated Berge formula, in which dual optimizers are known as barriers. Several results regarding maximal barriers have been derived by known canonical decompositions, however no characterization has been known for general graphs. In this paper, we provide a characterization of the family of maximal barriers in general graphs, in which the known results are developed and unified

Popular matchings with two-sided preferences and one-sided ties

We are given a bipartite graph $G = (A \cup B, E)$ where each vertex has a preference list ranking its neighbors: in particular, every $a \in A$ ranks its neighbors in a strict order of preference, whereas the preference lists of $b \in B$ may contain ties. A matching $M$ is popular if there is no matching $M'$ such that the number of vertices that prefer $M'$ to $M$ exceeds the number of vertices that prefer $M$ to~$M'$. We show that the problem of deciding whether $G$ admits a popular matching or not is NP-hard. This is the case even when every $b \in B$ either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each $b \in B$ puts all its neighbors into a single tie. That is, all neighbors of $b$ are tied in $b$'s list and $b$ desires to be matched to any of them. Our main result is an $O(n^2)$ algorithm (where $n = |A \cup B|$) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in $B$ have no preferences and do not care whether they are matched or not.Comment: A shortened version of this paper has appeared at ICALP 201

Reset thresholds of automata with two cycle lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We describe reset thresholds of the colorings of primitive digraphs with exactly one cycle of length p and one cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one cycle of length q and two cycles of length p.Comment: 11 pages, 5 figures, submitted to CIAA 201

Matching structure and bargaining outcomes in buyer–seller networks

We examine the relationship between the matching structure of a bipartite (buyer-seller) network and the (expected) shares of the unit surplus that each connected pair in this network can create. We show that in different bargaining environments, these shares are closely related to the Gallai-Edmonds Structure Theorem. This theorem characterizes the structure of maximum matchings in an undirected graph. We show that the relationship between the (expected) shares and the tructure Theorem is not an artefact of a particular bargaining mechanism or trade centralization. However, this relationship does not necessarily generalize to non-bipartite networks or to networks with heterogeneous link values

Staphylococcal enterotoxin A (SEA) stimulates STAT3 activation and IL-17 expression in cutaneous T-cell lymphoma

Cutaneous T-cell lymphoma (CTCL) is characterized by proliferation of malignant T cells in a chronic inflammatory environment. With disease progression, bacteria colonize the compromised skin barrier and half of CTCL patients die of infection rather than from direct organ involvement by the malignancy. Clinical data indicate that bacteria play a direct role in disease progression, but little is known about the mechanisms involved. Here, we demonstrate that bacterial isolates containing staphylococcal enterotoxin A (SEA) from the affected skin of CTCL patients, as well as recombinant SEA, stimulate activation of signal transducer and activator of transcription 3 (STAT3) and upregulation of interleukin (IL)-17 in immortalized and primary patient-derived malignant and nonmalignant T cells. Importantly, SEA induces STAT3 activation and IL-17 expression in malignant T cells when cocultured with nonmalignant T cells, indicating an indirect mode of action. In accordance, malignant T cells expressing an SEA-nonresponsive T-cell receptor variable region β chain are nonresponsive to SEA in monoculture but display strong STAT3 activation and IL-17 expression in cocultures with SEA-responsive nonmalignant T cells. The response is induced via IL-2 receptor common γ chain cytokines and a Janus kinase 3 (JAK3)-dependent pathway in malignant T cells, and blocked by tofacitinib, a clinical-grade JAK3 inhibitor. In conclusion, we demonstrate that SEA induces cell cross talk-dependent activation of STAT3 and expression of IL-17 in malignant T cells, suggesting a mechanism whereby SEA-producing bacteria promote activation of an established oncogenic pathway previously implicated in carcinogenesis

Malignant inflammation in cutaneous T-cell lymphoma: a hostile takeover

Cutaneous T-cell lymphomas (CTCL) are characterized by the presence of chronically inflamed skin lesions containing malignant T cells. Early disease presents as limited skin patches or plaques and exhibits an indolent behavior. For many patients, the disease never progresses beyond this stage, but in approximately one third of patients, the disease becomes progressive, and the skin lesions start to expand and evolve. Eventually, overt tumors develop and the malignant T cells may disseminate to the blood, lymph nodes, bone marrow, and visceral organs, often with a fatal outcome. The transition from early indolent to progressive and advanced disease is accompanied by a significant shift in the nature of the tumor-associated inflammation. This shift does not appear to be an epiphenomenon but rather a critical step in disease progression. Emerging evidence supports that the malignant T cells take control of the inflammatory environment, suppressing cellular immunity and anti-tumor responses while promoting a chronic inflammatory milieu that fuels their own expansion. Here, we review the inflammatory changes associated with disease progression in CTCL and point to their wider relevance in other cancer contexts. We further define the term "malignant inflammation" as a pro-tumorigenic inflammatory environment orchestrated by the tumor cells and discuss some of the mechanisms driving the development of malignant inflammation in CTCL