Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality among all total dominating sets of $D$. Given an undirected graph $G$, we study the maximum and minimum total domination numbers among all orientations of $G$. That is, we study the upper (or lower) orientable domination number of $G$, $\rm{DOM}_t(G)$ (or $\rm{dom}_t(G)$), which is the largest (or smallest) total domination number over all orientations of $G$. We characterize those graphs with $\rm{DOM}_t(G) =\rm{dom}_t(G)$ when the girth is at least $7$ as well as those graphs with $\rm{dom}_t(G) = |V(G)|-1$. We also consider how these parameters are effected by removing a vertex from $G$, give exact values of $\rm{DOM}_t(K_{m,n})$ and $\rm{dom}_t(K_{m,n})$ and bound these parameters when $G$ is a grid graph.

We introduce, and partially resolve, a conjecture that brings a three-centuries-old derangements phenomenon and its much younger two-decades-old analogue under the same umbrella. Our tools blend combinatorics and analysis in a medley incorporating Inclusion-Exclusion and Tannery’s theorem.

We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Tur\'an numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstra\ete. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\geq 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the traditional extremal number. We also show that saturation numbers are linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of K\'asonyi and Tuza. We conclude this paper with related open questions and conjectures.

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $\mathcal{C}$. Similarly to classical saturation functions, define $\mathrm{sat}_t(n, \mathcal{C})$ to be the minimum number of edges in a $(\mathcal{C},t)$ saturated graph. Let $\mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for $\mathrm{sat}_t(n, \mathcal{C}_r(K_k))$ for all $r$, showing a sharp change in behavior when $r\geq \binom{k-1}{2}+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine $\mathrm{sat}_t(n, \mathcal{C}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Turan number of $F$. We show that $ex^*(n,P_k)\geq \frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\leq 10$ and a few other caterpillars of small diameter.

BACKGROUND:The Food and Drug Administration recommends that retrievable inferior vena cava filters (IVCFs) be removed 29-54 days postinsertion. Nationally, the retrieval rate is around 23-25%. The objectives of this study are to assess the effect of a plan for IVCF retrieval and access to an office endovascular center (OEC) on filter removal rates and to assess the safety of the procedure in an OEC. METHODS:In this institutional review board-exempt retrospective study, the medical records of all patients who had an IVCF placed or removed in the hospital and OEC setting by one group of vascular surgeons between January 2011 and February 2017 were analyzed. Informed consent was not required for this retrospective chart review. The following data were abstracted: filter model, procedure site, retrieval plan, number of removal attempts, complications attributed to removal, success of removal, and the duration that the filter was in place. Anticoagulation was not discontinued before filter retrieval. Filters were removed under local anesthesia with or without mild conscious sedation. RESULTS:IVCF removal was attempted in all eligible patients, 120 of 191 with IVCFs, whereas 71 patients were lost to follow-up (46), died (19), or the indication changed (6). Of the patients who had filters placed in the hospital (n = 161), 62% were removed (n = 101), of which 86% had a removal attempt in the OEC, whereas 14% had the filter removed in the hospital. Sixty-three percent of patients who had filters placed in the OEC (n = 30) had the filter removed in the OEC (n = 19). All patients with a newly placed filter were given an office appointment with a vascular surgeon for evaluation and removal planning. Of patients who had their filter removed at the OEC, all were removed via the jugular approach, resulting in 103 of 106 (97%) successful removals in the OEC. Visipaque (GE Healthcare, Chicago, IL) contrast was used during filter removal. Intravascular ultrasound was not used because the study predates the insurance coverage of this technology in the office laboratory. There was no mortality related to filter removal. In addition, there were no bleeding complications, despite patients remaining on anticoagulant therapy during the removal. In 4% of patients, the filter was removed in less than 3 weeks, 30% of patients between 3 and 6 weeks, 26% of patients between 6 weeks and 3 months, and 40% of patients after 3 months. CONCLUSIONS:Having access to both an OEC and a documented retrieval plan increases the frequency of IVCF removal in a community compared with national rates. Retrievable filters can be safely removed in an OEC with extremely high success and safety. Anticoagulation therapy can be continued during retrieval attempt without increased risk of bleeding.

Objective: To evaluate the safety and effectiveness of infrainguinal artery revascularization via atherectomy supplemented with other endovascular techniques in an office endovascular center (OEC) setting. Methods: A retrospective study was conducted examining 352 lower extremity atherectomy revascularization procedures between 2011 and 2016 at an OEC by five board-certified vascular surgeons. Patients received laser atherectomy or orbital atherectomy followed by angioplasty or angioplasty and stent placement as needed. Reintervention was indicated based on evidence of clinical symptoms and imaging studies. Demographics, vessel-specific data, treatment information, and outcome of procedures were recorded. Data analysis was carried out using Kaplan-Meier survival curves. Results: Lower extremity atherectomy was carried out in 282 patients in 352 limbs with average age of 69 +/- 11 years. Technical success of <30% residual stenosis by angiogram was achieved in 571/594 vessels treated. Within 30 days of procedure, 23/352 limbs required major amputation resulting from pre-existing disease, ranging from 3 Rutherford class 4, 17 Rutherford class 5, to 3 Rutherford class 6 limbs. No 30-day mortality was noted. The primary patency of the 571 treated vessels at 12 months was 90%, and 84% at 29 months. The patency of treated vessels that reached >50% stenosis on follow-up and required reintervention (51/571 vessels) or did not require reintervention (79/571) was 72% and 87% at 23 months' follow-up, respectively, with no difference in risk of occlusion identified (P = .181). There was a significantly increased risk of occlusion for vessels treated with laser atherectomy as compared with orbital atherectomy (odds ratio, 2.552; 95% confidence interval, 1.375-4.735; P=.003). No significant difference in risk of occlusion was found between treatment with atherectomy and angioplasty (466/571 vessels) compared with atherectomy, angioplasty, and stenting (102/571) with secondary patency of 90% and 85% at 6 months' follow-up, respectively. There was no difference in patency between claudicants and patients with critical limb ischemia. Conclusions: Atherectomy in conjunction with angioplasty and/or stenting has satisfactory patency with minimal complications when the procedure is carried out in an OEC. Asymptomatic >50% restenosis of treated vessels does not warrant reintervention unless the patient presents with clinical symptoms. Various atherectomy devices may result in different outcomes.

OBJECTIVE:The objective of this study was to evaluate the safety and efficacy of a retrograde tibial approach in revascularization of lower extremities for treatment of ischemia in anatomically challenging patients. METHODS:This is a retrospective study of 57 procedures performed between 2012 and 2016 using the retrograde approach to treat patients with flush occlusion, inability to cross the lesion, failed bypass, or hostile groin. Demographic data, Rutherford classes, vessels treated, and approach were noted. Type of procedure, complications, amputations, deaths, and patency of access tibial vessels and treated vessels were recorded. Ultrasound-guided tibial access was achieved through the anterior tibial artery, posterior tibial artery, or peroneal artery. Technical success was defined as residual stenosis of <30%. Restenosis was defined as two times increase in velocity at the site of treatment. In follow-up, access vessel patency and treated vessel patency were evaluated by physical examination and ultrasound. Kaplan-Meier survival curves were used to assess proportional hazards before using the marginal Cox model to determine statistical significance in risk of postintervention occlusion. RESULTS:In 53 patients (32 men) with an average age of 67 ± 10.6 years, Rutherford categories were as follows: class 2, n = 1; class 3, n = 37; class 4, n = 5; class 5, n = 12; and class 6, n = 2. Tibial arteries were successfully accessed in all limbs. Lesions were crossed in 56 of 57 limbs. One procedure was terminated because of local arterial dissection. Revascularization was achieved in 55 of 57 limbs. Within 30 days of the procedure, 2 of 2 Rutherford class 6 patients and 1 of 12 class 5 patients needed major amputation because of pre-existing disease. There was no 30-day mortality. Of 103 vessels treated, technical success was achieved in 97 (94%). Secondary patency for 103 vessels was 79% with mean follow-up of 6.66 ± 5.4 months. The primary patency was 90% compared with a primary assisted patency of 51%. There was no statistically significant difference in access vessel primary patency in follow-up: 86% (30/35) for anterior tibial artery, 80% (16/20) for posterior tibial artery, and 100% (2/2) for peroneal artery. In addition, in follow-up, there was no significant difference in incidence of occlusion of target vessels based on choice of access vessel used (P = .109). CONCLUSIONS:In this group of anatomically challenging patients, a retrograde tibial approach was safely used. Accessing the tibial artery does not usually cause access vessel occlusion and resulted in no adverse outcomes. The majority of access vessels remained patent for future bypass if necessary.

For bipartite graphs F and H with Ramsey number R(F, H) = n and an integer k with 2 <= k <= n, the k-Ramsey number of F and H is the minimum order of a balanced complete k-partite graph G for which every red-blue coloring of G results in a sub graph of G isomorphic to F all of whose edges are colored red or a subgraph isomorphic to H all of whose edges are colored blue. In this work, we investigate the k-Ramsey numbers R-k(F, H) for certain stripes F and H (1-regular graphs) and for certain values of k. We also include a discussion of k-Ramsey numbers of graphs that are not bipartite.

The Food and Drug Administration recommends that retrievable inferior vena cava filters (IVCF) be removed between 25 and 54 days post insertion. Nationally, the retrieval rate is abysmally low. The objectives of this study are to (1) assess if a retrieval plan and access to an office endovascular center (OEC) will increase retrieval rates, and (2) assess the safety of the procedure in an OEC. In this institution review board-approved retrospective study, the medical records of patients who had an IVCF placed and/or removed in the hospital and OEC setting between January 2011 and January 2017 were analyzed. Demographic data collected include: hypercoagulable status, indication for insertion, point of entry, filter model, procedure site, documented retrieval plan, number of removal attempts, point of entry for removal, complications attributed to removal, success of removal, and the duration that the filter was in place. Anticoagulation was continued before filter retrieval. Filters were removed under local anesthesia or mild conscious sedation. IVCF retrieval was attempted in 116 of 214 patients, while the rest were lost to follow-up, died, or the indication changed. Of the patients who had filters placed in the hospital, 76% (n = 83) had a removal attempt in the OEC, while 14% had the filter removed in the hospital. Patients who had IVCFs inserted in an OEC had their filters removed in the OEC only. A documented retrieval plan was found in the medical record of 95% of patients. All filters were removed via the jugular approach, resulting in 99 of 101 successful removals (98%) in the OEC. There was no mortality related to filter removal. Additionally, there were no bleeding complications, despite the fact that patients remained on anticoagulant therapy during the removal. In 4% of patients, the filter was removed in less than 3 weeks, 29% of patients between 3 and 6 weeks, 26% of patients between 6 weeks and 3 months, and 40% of patients after 3 months. The most common filter removed was the Celect Cook (42.3%), followed by Eclipse Bard (18.8%), and Denali Bard (17.9%). Various other filters made up the remaining 21% (Fig). The incidence of removal of retrievable filters improves with a well-documented removal plan. Access to an OEC further helps increase the frequency of removal because of ease of access. Furthermore, retrievable filters can be safely removed in an OEC with extremely high success rates (98%). Finally, anticoagulation therapy can be continued during retrieval attempt without untoward complications.

A balanced complete k-partite graph, k >= 2, is a complete k-partite graph the degrees of whose vertices differ by at most 1. For graphs F and H and an integer k with 2 <= k <= R(F, H), where R(F, H) is the Ramsey number of F and H, the k-Ramsey number R-k(F, H) of F and H, if it exists, is the smallest order of a balanced complete k-partite graph G for which every red blue coloring of the edges of G results in a red F or a blue H. For an integer t >= 3, the unicyclic-star graph U-t is the unicyclic graph containing the star K1,t as a spanning subgraph. The k-Ramsey numbers R-k(U-t) = R-k(U-t, U-t) are determined for many pairs k, t of positive integers.

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a {\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the {\emph rainbow Tur\'an number} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete in 2007. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $k$ edges, disproving a conjecture in Keevash et al.

This study evaluated the safety and efficacy of retrograde tibial approach in revascularization of lower extremity for treating ischemia in anatomically challenging patients. In this Institutional Review Board-approved retrospective study, 56 patients underwent 60 procedures between 2012 and 2016, during which 112 vessels were treated using a retrograde approach due to flush occlusion, inability to cross lesion, failed bypass, or hostile groin. Demographic data, Rutherford classes, vessels treated, and vessel approach were noted. Type of procedure, complications, amputations, deaths, and patency of access tibial vessels and treated vessels were recorded. Technical success was defined as residual stenosis of <50%. Restenosis was defined as a two-times increase in velocity at the site of treatment. In follow-up, access and treated vessel patency were evaluated by physical examination and ultrasound imaging. Life-table analysis was performed. The χ2 test was used to test relations between risk factors and occlusion. There were 56 patients (34 males). Average age was 67 ± 11.3 years. Rutherford categories were class II (n = 1), class III (n = 39), class IV (n = 5), class V (n = 13), and class VI (n = 2). The lesion was crossed in 59 of 60 limbs. One procedure was halted due to local dissection. Within 30 days of procedure, two of two Rutherford class VI and one of 13 class V patients needed major amputation. There was no 30-day mortality. Technical success occurred in 105 of 112 vessels (93.8%). In follow-up, overall patency for 112 vessels at 19 months was 53.98%, 83 of 112 vessels (74.1%) not requiring secondary intervention had a patency of 75.4% at 19 months, and 29 of 112 vessels (25.9%) needing reintervention had a patency of 20.3% at 17 months (Fig 1). During follow-up, 32 of 37 (86.5%) of the AT, 19 of 21 (90.5%) of PT, and 2 of 2 (100%) of peroneal access vessels remained patent. Effect of various factors on vessel outcome is listed in the Table. In this anatomically challenging patient group, the tibial approach can be safely used with good medium term results. The retrograde approach rarely causes access vessel occlusion and results in no adverse outcome. Access vessel is not compromised for future bypass. Patients in Rutherford class II to III do better than those in IV to VI. Postprocedure anticoagulant treatment, excluding clopidogrel and warfarin, has a beneficial effect in improving follow-up patency.TableEffect of various factors on target vessel occlusion post interventionFactorsOR (95% CI)P valueRutherford Group (II-III vs IV-VI)2.94 (1.32-6.67).008Antiplatelet pre-op2.27 (0.92-5.59).07Antiplatelet post-op1.09 (0.44-2.68).86Warfarin pre-op0.53 (0.11-2.50).89Warfarin post-op0.35 (0.06-1.99).95Clopidogrel pre-op1.34 (0.57-3.14).50Clopidogrel post-op2.07 (0.81-5.28).12Other anticoagulants pre-op0.72 (0.17-3.04).79Other anticoagulants post-op0.14 (0.04-0.54).002aCI, Confidence interval; OR, odds ratio.aStatistically significant. Open table in a new tab

This study determined the safety and efficacy of infrainguinal artery atherectomy. In this Institutional Review Board-approved retrospective study, 260 patients underwent orbital or laser atherectomy on 362 limbs in 615 vessels between 2011 and 2016. Demographic data, Rutherford class, vessels treated, vessel approach, and type of device used were noted. Success and type of procedure, complications, 30-day major amputations, and deaths were recorded. Technical success was defined as <50% stenosis. Restenosis was defined as a two-times increase in velocity at the treated site. In follow-up, vessel patency was evaluated by examination and ultrasound imaging. Technical success was defined by residual stenosis <50%. Symptomatic patients with restenosis of >50% underwent reintervention, and asymptomatic patients did not undergo reintervention. Life-table analyses were performed. The χ2 test was used to test relations between risk factors and occlusion. Rutherford class distribution was 2 (n = 8), 3 (n = 166), 4 (n = 41), 5 (n = 103), and 6 (n = 11). There were 23 major amputations on 362 limbs (6%) in Rutherford class 4 (3 of 41), 5 (17 of 103), and 6 (3 of 11). Mortality was 0. Complications included perforation in 3, abrupt closure in 2, and embolization in 1. Technical success was 604 of 615 vessels (98%). By atherectomy device, orbital (n = 515) patency was 91% (mean follow-up, 29 ± 31 months) and significantly better than laser (n = 100) patency at 78% (mean follow-up, 33 ± 32 months). Previous bypass had a significant negative impact (P = .04). Preoperative clopidogrel (P = .04) or preoperative (P = .01) and postoperative (P = .003) anticoagulants had a significant salutary effect. Patency after angioplasty was 89% in 498 vessels, with a mean follow-up of 31 ± 31 months. Patency in 106 vessels treated with angioplasty and stent was 86%, with a mean follow–up of 23 ± 29 months. After secondary intervention, vessels had lower patency (Fig 1). Patients with >50% stenosis who had secondary intervention had worse results than patients who did not (Fig 2). In medium-term follow-up, excellent patency is achieved using atherectomy and angioplasty with minimal complications. Asymptomatic restenosis of >50% does not warrant reintervention. These procedures have comparable results among various Rutherford classes. Use of anticoagulants is beneficial. Various atherectomy devices may result in different outcomes.Fig 2Kaplan-Meier curves for vessel survival by patency group starting at 50% stenosis. Stenosis >50% developed in 143 vessels (23.3%); of those, 91 (64%) had no intervention and a 96% patency at a mean follow-up 30 ± 28 months. Fifty-two (36%) received reintervention and had a patency of 67%, with a mean follow-up of 35 ± 33 months.View Large Image Figure ViewerDownload Hi-res image Download (PPT)

Radial artery access for percutaneous interventions was first introduced in 1989 for diagnostic coronary angiography. Radial access is being used in about 2% of peripheral percutaneous cases nationally, while femoral approach remains the preferred method. We present our experience with radial artery access for diagnosis of peripheral artery disease and intervention in our office endovascular lab.