Let \(\operatorname{PW}(G)\) and \(\operatorname{TW}(G)\) denote the path-width and tree-width of a graph \(G\), respectively. Let \(G+H\) denote the join of two graphs \(G\) and \(H\). We show in this paper that

\(\operatorname{PW}(G + H) = \min\{|V(G)| + \operatorname{PW}(H),|V(H)| + \operatorname{PW}(G)\}\)

and

\(\operatorname{TW}(G + H) = \min\{|V(G)| + \operatorname{TW}(H), |V(H)| + \operatorname{TW}(G)\}\).

For a positive integer \(k\), a \(k\)-subdominating function of \(G = (V, E)\) is a function \(f: V \to \{-1, 1\}\) such that the sum of the function values, taken over closed neighborhoods of vertices, is at least one for at least \(k\) vertices of \(G\). The sum of the function values taken over all vertices is called the aggregate of \(f\) and the minimum aggregate amongst all \(k\)-subdominating functions of \(G\) is the \(k\)-subdomination number \(\gamma_{ks}(G)\). In the special cases where \(k = |V|\) and \(k = \lceil|V|/2\rceil\), \(\gamma_{ks}\) is respectively the signed domination number [{4}] and the majority domination number [{2}]. In this paper we characterize minimal \(k\)-subdominating functions. By determining \(\gamma_{ks}\) for paths, we give a sharp lower bound for \(\gamma_{ks}\) for trees. We also determine an upper bound for \(\gamma_{ks}\) for trees which is sharp for \(k \leq |V|/2 \).

Let \(G\) be a connected (multi)graph. We define the leaf-exchange spanning tree graph \( {T_l}\) of \(G\) as the graph with vertex set \(V_l = \{T|T \text{ is a spanning tree of } G\}\) and edge set \(E_l = \{(T, T’)|E(T)\Delta E(T’) = \{e, f\}, e \in E(T), f \in E(T’) \text{ and } e \text{ and } f \text{ are incident with a vertex } v \text{ of degree } 1 \text{ in } T \text{ and } T’\}\). \({T}(G)\) is a spanning subgraph of the so-called spanning tree graph of \(G\), and of the adjacency spanning tree graph of \(G\), which were studied by several authors. A variation on the leaf-exchange spanning tree graph appeared in recent work on basis graphs of branching greedoids. We characterize the graphs which have a connected leaf-exchange spanning tree graph and give a lower bound on the connectivity of \( {T_l}(G)\) for a \(3\)-connected graph \(G\).

The fine structure of a directed triple system of index \(\lambda\) is the vector \((c_1,c_2,\ldots,c_\lambda)\), where \(c_i\) is the number of directed triples appearing precisely \(i\) times in the system. We determine necessary and sufficient conditions for a vector to be the fine structure of a directed triple system of index \(3\) for \(v \equiv 0\) or \(1 \pmod{3}\).

Let \(p\) denote the circumference of a two-connected graph \(G\). We construct a hamiltonian cycle in \(G^2\) which contains more than \(p/2\) edges of \(G\). Using this construction we prove some properties of hamiltonian cycles in the square of \(G\).

For a connected graph \(G\) that is not a cycle, a path or a claw, let its \(k\)-iterated line graph have the diameter \(diam_k\), and the radius \(r_k\). Then \(diam_{k+1} = diam_k + 1\) for sufficiently large \(k\). Moreover, \(\{r_k\}\) also tends to infinity and the sequence \(\{diam_k – r_k – \sqrt{2\log_2 k}\}\) is bounded.

In \([1]\) it is proved that each \(4\)-critical plane graph contains either a \(4\)- or a \(5\)-cycle or otherwise a face of size between \(6\) and \(11\).

For nonempty graphs \(G\) and \(H\), \(H\) is said to be \(G\)-decomposable (written \(G|H\)) if \(E(H)\) can be partitioned into sets \(E_1, \ldots, E_n\) such that the subgraph induced by each \(E_i\) is isomorphic to \(G\). If \(H\) is a graph of minimum size such that \(F|H\) and \(G|H\), then \(H\) is called a least common multiple of \(F\) and \(G\). The size of such a least common multiple is denoted by \(\mathrm{lcm}(F,G)\). We show that if \(F\) and \(G\) are bipartite, then \(\mathrm{lcm}(F,G) \leq |V(F)|\cdot|V(G)|\), where equality holds if \((|V(F)|,|V(G)|) = 1\). We also determine \(\mathrm{lcm}(F,G)\) exactly if \(F\) and \(G\) are cycles or if \(F = P_m, G = K_n\), where \(n\) is odd and \((m-1,\frac{1}{2}(n-1)) = 1\), in the latter case extending a result in [{8}].

Let \(G\) be a graph. A vertex subversion strategy of \(G\), \(S\), is a set of vertices in \(G\) whose closed neighborhood is deleted from \(G\). The survival-subgraph is denoted by \(G/S\). The vertex-neighbor-integrity of \(G\), \(\mathrm{VNI}(G)\), is defined to be \(\mathrm{VNI}(G) = \displaystyle\min_{S\subseteq V(G)} \{|S| + w(G/S)\}\), where \(S\) is any vertex subversion strategy of \(G\), and \(w(G/S)\) is the maximum order of the components of \(G/S\). In this paper, we show the minimum and the maximum vertex-neighbor-integrity among all trees with any fixed order, and also show that for any integer \(l\) between the extreme values there is a tree with the vertex-neighbor-integrity \(l\).