A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function f is the sum, u∈V (G) f (u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.2010 Mathematics Subject Classification. 05C69.

Epidermodysplasia verruciformis (EV) is a rare dermatologic condition that is clinically characterized by flat, cutaneous, verrucous papules, pityriasis versicolor-like lesions, and similar lichenoid papules. There are 2 forms of EV: a classic inherited genodermatosis and a secondary acquired form. EV predisposes individuals to infections with certain types of human papillomavirus virus and subsequently increases the risk of cutaneous squamous cell carcinoma. The acquired form occurs in immunosuppressed patients, particularly in patients infected with HIV; however, it has also been described in patients who have undergone stem cell and solid organ transplantation. We report an additional case of renal transplantation and immunosuppressive therapy-associated acquired EV (AEV) in a 78-year-old man with multiple flesh-colored to violaceous, flattopped papules distributed on the face and trunk clinically mimicking lichen planus. Biopsy was typical for that of EV, demonstrating enlarged keratinocytes with a blue-gray cytoplasm, a thickened granular layer, acanthosis, and hyperkeratosis. Herein, we discuss an unusual presentation of an AEV-mimicking lichen planus with review of the literature.

In this paper, we continue the study of the L-Grundy domination number of a graph introduced and first studied in [Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017) 66-77]. A vertex in a graph dominates itself and all vertices adjacent to it, while a vertex totally dominates another vertex if they are adjacent. A sequence of distinct vertices in a graph G is called an L-sequence if every vertex v in the sequence is such that v dominates at least one vertex that is not totally dominated by any vertex that precedes v in the sequence. The maximum length of such a sequence is called the L-Grundy domination number, L gr(G), of G. We show that the L-Grundy domination number of every forest G on n vertices equals n, and we provide a linear-time algorithm to find an L-sequence of length n in G. We prove that the decision problem to determine if the L-Grundy domination number of a split graph G is at least k for a given integer k is NP-complete. We establish a lower bound on L gr(G) when G is a regular graph, and investigate graphs G on n vertices for which L gr(G) = n.

Histologic distinction between melanoma ex-blue nevus and cellular blue nevus (CBN) can often be difficult, but features supporting melanoma include infiltrative growth pattern, frequent mitoses, cytologic atypia and pleomorphism, cell crowding, and tumor necrosis. Unfortunately, these features are not constantly dependable and frequently borderline lesions exist, so-called atypical CBN, which lack explicit malignant features. Furthermore, some CBN and atypical CBN show an assortment of features, which may lead to their misdiagnosis as melanoma, but to date necrosis is generally absent. We present an unusual case of an atypical cellular blue nevus with extensive necrosis mimicking melanoma ex-blue nevus.

De‐differentiated chondrosarcoma (DDCS) is an extremely aggressive tumor of the bone characterized by a high‐grade, non‐chondroid sarcoma adjacent to a low‐ or intermediate‐grade chondrosarcoma. Adequate tumor sampling demonstrating the biphasic features is necessary to make an accurate diagnosis. The diagnosis may be challenging as histopathology may mimic other neoplasms. We present a case of a 76‐year‐old woman with a history of breast cancer who presented with a pathologic non‐displaced fracture. A bone biopsy demonstrated a high‐grade neoplasm composed of pleomorphic spindled and epithelioid cells with focal expression of AE1/3 and GATA3, most likely consistent with metastatic breast carcinoma. After a difficult clinical course, the tumor was resected demonstrating a similar morphology to her prior biopsy, as well as an area of a low‐grade cartilaginous neoplasm consistent with chondrosarcoma. The biphasic tumor alongside a low‐grade chondrosarcoma allowed for a diagnosis of DDCS. Several days after her procedure, the patient developed violaceous nodules overlying and surrounding the surgical site. Skin biopsy demonstrated a malignant epithelioid neoplasm with identical histomorphologic features identical to her prior bone resection. Given the location of the skin lesions directly within the surgical site right after resection, the clinical‐pathological picture was that of sarcomatosis cutis by iatrogenic cutaneous implantation.

Talimogene laherparepvec (TVEC) is a genetically modified herpes simplex virus-1 approved as an intralesional oncolytic immunotherapy for the treatment of advanced melanoma. Cutaneous reactions at the site of injection may mimic recurrent or progressive melanoma; histopathological findings have included chronic granulomatous dermatitis, neutrophilic dermatitis, lymphocytic dermatitis, and pigment incontinence. We report a 39-year-old male with metastatic stage IIIc melanoma treated with TVEC with clinical regression of melanoma lesions that later developed pink nodules at sites of prior injection. Histopathology demonstrated a nodular mononuclear infiltrate that stained strongly and diffusely with CD45 and CD20 with a surrounding rim of CD3-positive T-cells. Immunoglobulin gene rearrangement was negative for a clonal B-cell population. To our knowledge, this is the first report of a pseudolymphomatous reaction mimicking recurrent melanoma after TVEC therapy.

A perfect Roman dominating function on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted ?pR(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a cubic graph on n vertices, then ?pR(G) ? 3/4n, and this bound is best possible. Further, we show that if G is a k-regular graph on n vertices with k at least 4, then ?pR(G) ? (k2+k+3/k2+3k+1)n.

Pyemotes ventricosus mites are an uncommon cause of pruritic dermatitis seen most commonly in occupational exposure, prominently found in professionals such as farmers, landscapers, and factory workers who work with grains, wheat, dried beans, or grasses. The clinical description of the rash has typically been described as papular, erythematous, with a central vesicular lesion. We describe a case of Pyemotes dermatitis with an atypical clinical presentation. A 30-year-old man presented with pruritic, umbilicated papules, which involved his right lateral trunk and upper thigh leading to the submitted clinical impression of molloscum contagiosum. A biopsy of the skin was taken, and fragments of arthropod consistent with P. ventricosus were identified within umbilicated indentations of skin. The patient subsequently admitted to the onset of the rash immediately after carrying bales of straw while supporting each bale with his right side. The possibility of Pyemotes dermatitis mimicking a poxvirus–like eruption should be considered when encountering an unusual umbilicated papular eruption in the appropriate patient with occupational exposure.